Computer Networks 54 (2010) 208–217

Contents lists available at ScienceDirect

Computer Networks journal homepage: www.elsevier.com/locate/comnet

Complexity results on labeled shortest path problems from wireless routing metrics Charles B. Ward a,*, Nathan M. Wiegand b a b

Department of Computer Science, Stony Brook University, Stony Brook, NY 11794-4400, United States Department of Computer Science, The University of Alabama, Box 870290, Tuscaloosa, AL 35487-0290, United States

a r t i c l e

i n f o

Article history: Available online 30 June 2009 Keywords: Wireless routing metrics Labeled paths Shortest paths Approximation algorithms

a b s t r a c t Metrics to assess the cost of paths through networks are critical to ensuring the efficiency of network routing. This is particularly true in multi-radio multi-hop wireless networks. Effective metrics for these networks must measure the cost of a wireless path based not only on traditional measures such as throughput, but also on the distribution of wireless channels used. In this paper, we argue that routing metrics over such networks may be viewed as a class of existing shortest path problems, the formal language constrained path problems. On this basis, we describe labeled path problems corresponding to two multi-radio wireless routing metrics: Weighted Cumulative Expected Transmission Time (WCETT), developed by Draves et al., and Metric for Interference and Channel-switch (MIC), developed by Yang et al. For the first, we give a concise proof that calculating shortest WCETT paths is strongly NP-Complete for a variety of graph classes. We also show that the existing heuristic given by Draves et al. is an approximator. For the second, we show that calculating loop-free (simple) shortest MIC paths is NP-Complete, and additionally show that the optimization version of the problem is NPO PB-Complete. This result implies that shortest simple MIC paths are only poorly approximable in the worst case. Furthermore, we demonstrate how the polynomial-time algorithm for shortest MIC paths is derivable from an existing language constrained shortest path algorithm. We use this as a basis to exhibit the general utility of viewing multi-channel wireless routing metrics as labeled graph problems, and discuss how a class of related polynomial-time computable metrics are derivable from this algorithm. Ó 2009 Elsevier B.V. All rights reserved.

1. Introduction Creating appropriate metrics to assess the cost of differing paths through a network is crucial for developing efficient routing algorithms. Such metrics should account for a variety of differences in the quality of paths, ranging from simple hop-count, to issues such as throughput and packet delay. Shortest path metrics which fail to do so risk creating routing paths which are inferior in practice. * Corresponding author. Tel.: +1 205 246 6686. E-mail addresses: [email protected] (C.B. Ward), nwiegand@cs. ua.edu (N.M. Wiegand). 1389-1286/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.comnet.2009.04.012

Although these factors are important in traditional wired networks, these differences are particularly acute in wireless networks. In situations where some or all links may be using wireless mediums, new difficulties arise, such as link interference (cross-talk between wireless channels). These difficulties are exacerbated by the wellknown problems in constructing wireless MAC protocols, such as with the implementation of collision detection. Because of such difficulties, the multi-radio multi-channel wireless network model has been introduced. This model allows individual wireless nodes to transmit over multiple radios using non-interfering channels. In this way, performance degrading interference may be reduced.

C.B. Ward, N.M. Wiegand / Computer Networks 54 (2010) 208–217

209

However, this model introduces new complexities to the problem of adequately assessing the cost of a routing path. Paths through a multi-radio, multi-hop wireless network consist not only of the intermediate wireless nodes but also the wireless channel assignments for transmission between them. Thus, in creating routing algorithms for such networks we have two major problems to consider. First, metrics must be developed which give realistic assessments of the effect of channel assignment on network transmissions. Second, algorithms which compute the most effective paths according to these metrics must be constructed. In this paper, we argue that calculating shortest paths over any metric which exploits channel assignment can be viewed as a shortest path problem over a multigraph with labeled edges. Specifically, we can relate these problems to work done on formal language constrained path problems. Typically, formal language constrained shortest path problems take as input a (weighted or unweighted) multigraph in which the edges have been labeled with terminal (alphabet) symbols from some formal language [1]. A path through the graph can then be associated with a string of alphabet symbols corresponding to the edges constituting the path. Thus, the shortest language constrained path is the shortest path through the graph with a corresponding string which is in the given formal language. Formally, we consider an all-pairs version of this problem:

Metric of Interference and Channel-switch to existing algorithms for labeled path problems studied by Barrett et al. [1], Yannakakis [3], and others. Furthermore, we demonstrate that the decision problem corresponding to calculating shortest paths over the Weighted Cumulative Expected Transmission Time [8] metric is Strongly NP-Complete on general directed and undirected graphs, directed acyclic graphs, and series-parallel graphs. We show the Strong NP-Completeness of computing shortest WCETT paths, which implies that there exists no pseudo-polynomial time algorithm for the problem. We also briefly discuss the provable approximability of this metric using the method given by Draves et al. [8]. We also show that calculating shortest simple paths over the Metric of Interference and Channel-switch [7] is NP-Complete over general directed graphs. We show that a slight generalization of the metric is NP-Complete for general directed and undirected graphs. Moreover, we show that the optimization version of both problems belongs to the NPO PB-Complete complexity class; that is, the most difficult NP optimization problems with objective functions polynomially bounded by the input size. This implies very poor approximability results for these problems, and implies that the MIC metric can theoretically have arbitrarily bad worst case behaviour when compared to the simple path if weighting constants are not chosen to mitigate this issue.

Definition 1. The all-pairs language constrained shortest path problem.

1.2. Prior work

Input: Finite set of symbols R, a Labeled Directed Multigraph Graph G ¼ ðV; EÞ where E # V  V  R with weight function w : E ! R, and formal language K over the alphabet R. Output: jVj  jVj matrix d such that dðu; vÞ ¼ minimum distance from u to v over any path such that the concatenation of the edge labels over the path is a valid string in K. This is a generalization of regular language and contextfree language constrained problems investigated by Mendelzon and Wood [2]; Yannakakis [3,4]; Barrett et al. [1]; and others [5,6]. While we will retain the above definition of a labeled multigraph throughout this paper, for notational convenience we will treat the labeling of edges as the output of a labeling function L : E ! R, rather than as the third term of the ordered triple. 1.1. Contributions In this paper, we discuss how routing metrics which take into account wireless channel assignment relate to the field of labeled graph problems. We argue that viewing wireless routing metrics in this light allows us to elegantly describe the calculation of many such metrics, and provides an existing base of knowledge which can be applied to such metrics. Specifically, we relate the polynomial algorithm given by Yang et al. [7] for shortest paths over

1.2.1. Routing metrics in wireless networks There is widespread interest in creating routing metrics which appropriately account for the complexities of multiradio multi-hop wireless networks. Proposed by De Couto et al. [9], Expected Transmission Count (ETX) is one early metric which has served as a basis for more recent metrics. In essence, ETX is the expected number of packet transmissions required to successfully deliver a packet across a path. It is argued that in practice, networks which minimize this metric will have high throughput. Performing significantly better than simple hop-count, ETX is also used as a component of several other metrics, including those we specifically examine in this paper. Draves et al. [8] extended the ETX metric, describing two metrics: Expected Transmission Time (ETT) and Weighted Cumulative Expected Transmission Time (WCETT). ETT extends ETX to incorporate the bandwidth of a link, while WCETT extends ETT by incorporating an additional constraint which works to spread a path over multiple wireless channels. We discuss these metrics in more detail in Section 2.1. Yang et al. [7] suggest the Metric for Interference and Channel-switch (MIC). Like WCETT, MIC uses ETT as a component of the metric, but the channel-specific component differs. MIC is concerned with the interference a flow may cause to itself when it is retransmitted through the same wireless channel as it arrived. Thus, MIC assigns higher costs to paths in which two consecutive hops of a path are transmitted over the same channel. We discuss MIC in more detail in Section 2.2.

210

C.B. Ward, N.M. Wiegand / Computer Networks 54 (2010) 208–217

Other metrics such as Exclusive Expected Transmission Time (EETT) and Adjusted Expected Transfer Delay (AETD) have been suggested [10,11]. Proposed by Jiang et al. [10], EETT also extends the concept of Expected Transmission Time, taking into account not only the rapidity of transfer over a link, but also the potential interference of neighboring links. Proposed by Zhou et al. [11], AETD uses an estimate of the jitter of links, and is designed to select a route in which links using the same wireless channel are separated ‘‘as far as possible.” 1.2.2. Formal language constrained path problems Formal language constrained path problems have found applications in a wide variety of subject areas throughout the field of computer science, from databases to constraint programming [2,3,6]. A number of general algorithms exist to calculate shortest paths over various types of languages, while NP-Completeness results have been shown for others. Below, we describe the applications to which language constrained path problems have been applied. We will discuss theoretical results from the area in Section 3. In one of the first papers to examine the problem of finding paths in labeled graphs, Mendelzon and Wood [2] examined simple labeled paths. They noted that many recursive database queries that arose in practice were easily and naturally expressible as graph traversals. Yannakakis also studied labeled path problems as they applied to database theory [3,4]. Specifically, because ‘‘relational databases can be viewed as labeled directed hypergraphs,” he examined labeled path problems as they applied to computing transitive closures in databases. In a similar vein, Abiteboul and Vianu examined navigational queries expressed in regular expressions [12]. These queries are useful for any number of contexts, ‘‘ranging from hypertext data to object-oriented databases” [12]. Barrett et al. have applied language constrained shortest path algorithms to transportation planning [13]. Barrett et al. have used these algorithms and greatly extended the theoretical literature [1] in their work on the TRANSIMS project, a transportation planning system supported by Los Alamos. Choppella and Haynes investigated the problem of source-tracking unification [6]. Unification has applications in a variety of areas such as automated theorem proving, artificial intelligence, programming languages, and logic programming. Non-unifiability has direct applications to type errors in other programming languages, as well [6,14,15]. Choppella and Haynes investigated the problem of deriving the shortest unification path for a given problem and determined that it is simply a shortest paths problem with a context-free language constraint. Bradford [16] considered a slightly different version of the context-free constrained shortest path problem, examining quickest labeled paths. Quickest paths were originally discussed by Moore [17] in reference to flowrateconstrained networks. Bradford extends the notion of quickest paths to include context-free language constraints and applies this notion to cryptographic routing.

1.3. Paper structure In Section 2 we consider the WCETT and MIC metrics in more detail, and in Section 3 we discuss the correspondence between shortest paths over these metrics and formal language-constrained path problems. In Sections 4 and 5 we describe labeled graph decision problems corresponding to calculating paths over the WCETT and MIC metrics and give our complexity results. Finally, we conclude the paper in Section 6. 2. Two routing metrics for wireless mesh networks In all networks, effectively evaluating the cost of using communications links is the basis for determining routing paths. In wireless networks particularly, however, accurately assessing the cost of a routing path is problematic. The inherent qualities of individual links must be accounted for, as should the interference effects that link usage causes on other links. These are particularly interesting problems in the case where individual routing nodes have multiple wireless radios which can be tuned to separate, non-interfering channels. We now describe two metrics which attempt to account for both of these factors to create a unified cost estimate for routing paths. Fig. 1 gives an example of a simple multi-radio multihop wireless network which we will use to elucidate our discussion of the WCETT and MIC metrics. In this figure, W represents the edge weight, while C represents the channels available to each node. Each node has two radios, tuned to two different channels. Edges simply represent radios within transmission range on the same channel, and the channels available between nodes are expressed as edge labels. 2.1. Weighted cumulative expected transmission time The first metric we will examine is WCETT, from the work of Draves et al. [8]. WCETT is designed to combine estimates of transmission time across links with channel information in wireless networks. WCETT consists of two components. The first is simply the sum of the ETT measures for each link along a given path. This measure generalizes the ETX metric created by De Couto, et al. [9]. ETT adjusts the ETX metric to take into account the bandwidth of the link being considered. Thus, for a given path, the sum of the ETT measures gives an estimate of the total transmission time of packets across the path. The second component of the WCETT metric is a measure designed to take wireless channel usage into account. This measure is the total ETT cost of the most used (by ETT) channel in the path. Thus, this term is generally minimized over channel diverse paths, though particularly high bandwidth and low loss channels may still take precedence over channel diversity. Formally, WCETT is defined as below, see [8]. Definition 2. Let p be a path in the network graph G ¼ ðV; EÞ, let C be a set of valid edge labels (wireless channels), let b be a tunable parameter in the range [0, 1], fixed for any instance of the problem, and let ETT : E ! Rþ

C.B. Ward, N.M. Wiegand / Computer Networks 54 (2010) 208–217

211

Fig. 1. Multi-radio multi-hop wireless network example.

be the weight function on the edges. Then the cost function WCETT is computed over p as follows:

WCETTðpÞ ¼ ð1  bÞ X j ðpÞ ¼

X

X e2p

ETTðeÞ þ b maxfX j ðpÞg; j2C

ETTðeÞ:

e2p labelðeÞ¼j

Taking Fig. 1 as an example, allow the W values to be the ETT values for each edge. Suppose that b ¼ 0:5. Then the shortest WCETT path from Node 1 to Node 5 is: 1 6 6 1 ! 2 ! 4 ! 5. This path has a cost of 6  0:5 from the first term, and 3  0:5 from the second, for a total cost of 4.5. The direct path from node 1 to node 5 through node 3, has a cost of 6 from each term, for a total cost of 6. However, if b ¼ 0, then these paths have equal cost of 6, and 6 6 6 the shortest path has a cost of 5: 1 ! 2 ! 4 ! 5. We will demonstrate the NP-Completeness of calculating shortest paths over this metric in Section 4. Draves, et al. compute an approximation of WCETT by using Dijkstra’s Algorithm [8]. We note that this is a jCj-approximator; that is, the solution will be within a multiplicative factor of jCj of the optimal solution. However, it is also true that the unlabeled shortest path (i.e., the shortest path by a traditional shortest path metric) is also a jCj-approximator. Lemma 3. The shortest unlabeled path is within a factor of jCj of the shortest WCETT path. Proof. Let pw be the shortest WCETT path and po be the shortest ETT path: ETTpo 6 ETTpw . Then the smallest possiwÞ , occurble value for WCETTðpw Þ is ð1  bÞETTðpw Þ þ b ETTðp jCj ring in the case where the labels split the edge weights perfectly evenly. The worst possible value for WCETTðpo Þ is ETTðpo Þ, occurring in the case where the edges along po

have only one label. Thus, the worst case is that b ¼ 1 and thus WCETTðpo Þ 6 jCjWCETTðpw Þ. h Lemma 4. The shortest path as computed by Dijkstra’s algorithm using WCETT distance as the heuristic function is within a factor of jCj of the shortest WCETT path. Proof. In order for the modified Dijkstra algorithm to fail to compute the shortest WCETT path, it must be that at some intermediate node x between the source s and the destination t there are two paths from s to x: (1) a path which is the shortest WCETT path from s to x but which does not fall on the shortest WCETT path from s to t, (2) a path which is not the shortest WCETT path from s to x but which does fall on the shortest WCETT path from s to t. Then, the worst possible case for the algorithm is that path (1) uses all channels equally to a level just under that of path (2), and that path (2) uses only one channel. Thus, path (2) essentially has a factor of jCj more potential to absorb channel cost in the remaining portion of the path from x to t. By repeating this on jCj intermediate vertices, the worst possible case of cost arbitrarily close to a factor of jCj times optimal can be achieved. It is never possible to do worse than this as, at each intermediate vertex, path (2) cannot beat path (1) in terms of WCETT cost. h When the number of channels is a small constant, this indicates that modified Dijkstra is a reasonable approximator. In the case of the 802.11b/g wireless standards, there are only 11 channels, only three of which can be non-overlapping simultaneously. However, this constant can also be fairly high. For example, the 802.11a wireless standard allows for up to 23 non-overlapping channels [18]. Furthermore, although the number of interfaces per node should be relatively small, due to antenna interference even on

212

C.B. Ward, N.M. Wiegand / Computer Networks 54 (2010) 208–217

non-overlapping channels, this does not affect the approximability of either algorithm. 2.2. Metric of interference and channel-switch The MIC metric for shortest paths in wireless mesh networks was created by Yang et al. [7], and was designed to improve upon WCETT. MIC consists of two principal components. The first component is the sum of individual weights on all links in the path. These weights are the IRU values: estimates of the total amount of wasted transmission time. That is, it is the expected transmission time multiplied by the number of links which interfere with the wireless channel being used. Thus, this component captures the potential for the path to interfere with other paths. The second component consists of a sum of weights for intermediate nodes in the path, where the weight is determined by the incoming and outgoing channels of the path at that node. That is, if two consecutive links on a path are transmitted on the same channel, then a flow along this path will interfere with its own transmission, reducing the viability of the path. Thus, the second term captures the potential for the path to interfere with itself. Formally, MIC is defined as below, see [7]. Definition 5. Let p; G; C; ETT, be defined as for WCETT, while minðETTÞ is the smallest edge weight in the graph. Then if v 2 V, let prevðv; pÞ be the node preceding v in path p or ; if there is no such node, and define Lð;Þ ¼ . Let N e be the set of neighboring edges which link e interferes with (has the same channel label). Finally, w1 ; w2 2 Rþ ; 0 6 w1 < w2 , are channel-switching costs. Then define the cost function MIC over p as follows:

MICðpÞ ¼

X X 1 IRU e þ CSC e;p ; jVj  minðETTÞ e2p e2p e2E

e2E

IRU e ¼ ETT e  Ne ;  w1 if Lðprevðv; pÞÞ – LðvÞ; CSC v;p ¼ w2 if Lðprevðv; pÞÞ ¼ LðvÞ: Let us take Fig. 1 again as an example; for simplicity allow the W values to be the scaled ETT values for each edges. First, let us take w1 ¼ 5 and w2 ¼ 13. In this case, 1 1 the shortest path from Nodes 1 to 5 is: 1 ! 3 ! 5. This path has a cost of 6 from the weights, and 13 from the channelswitching cost, for a total cost of 19. If we reduce w1 to 1, 6 1 11 6 however, the path 1 ! 2 ! 3 ! 4 ! 5 becomes the shortest, having a total weight of only 18, 15 from weights and 3 from channel-switching. Yang et al. give an algorithm that computes shortest paths according to this metric. This algorithm does this by augmenting the graph, such that each vertex in the original graph is replaced with a number of vertices equal to twice the number of edges at that vertex. Thus by connecting these vertices with directed edges with appropriate weights, the CSC cost can be enforced. It is important to note that a shortest path according to this metric may have a loop, as it may be possible to circumvent the increased cost of utilizing two edges in a

row with the same channel by leaving a vertex and returning across a different channel. However, such a loop creates only a faux savings, as the disruption to the transmission will, in practice, be the same, but with the addition of wasted bandwidth in the loop. Unfortunately, a shortest path according to the metric may not be the shortest loop-free path according to the metric, so although we may remove the loop in practice, this does not necessarily result in a shortest path according to the metric. It is arguable that this is actually a practical problem, of course, as this problem can only occur when the w2  w1 cost exceeds the IRU cost of the loop in question. This does, however, reinforce the importance of choosing good CSC values. Moreover, from a theoretical standpoint, we will show in this paper that for directed graphs there is very little to be done to correct this problem. As we will demonstrate, for directed graphs it is not possible to calculate shortest simple paths over MIC in polynomial time (assuming P – NP). Additionally, we show that a slight generalization, in which we allow arbitrary pairings of conflicting channels, is NP-Complete for directed and undirected graphs. Thus, our results imply that for this metric mitigation of the problem through choice of appropriate CSC values is the only practical solution.

3. Routing metrics and formal language constrained path problems The formal-language-constrained path problems have been studied by a number of authors, most recently and in the most depth by Barrett et al. [1]. These problems are concerned with finding shortest paths over graphs wherein the set of valid paths are limited to those which are labeled with strings from some formal language (given either as an input to the problem, or fixed as a constant). Edges in the graph are labeled with symbols from some alphabet R and the string corresponding to a path is simply taken as the concatenation of the labels of the edges in the path. It has been shown that these shortest path problems are computable in polynomial time for context-free languages (and hence all languages falling below in the Chomsky hierarchy), with a variety of algorithms specialized to the type of input language. These problems also become NP-Complete when restricted to simple paths, even for extremely simple classes of input languages. As an example, let us again consider Fig. 1. We may consider wireless channels to be alphabet symbols, and the channels of the links themselves as edge labels. Suppose, then, we wish to calculate a shortest path from Nodes 1 to 5, with the constraint that no label (channel) may be used on two consecutive links. Under this constraint, there are only three potential paths, all of which are forced to cross over the expensive edge between nodes 3 and 4. The short1 11 6 est of these is 1 ! 3 ! 4 ! 5, with a total cost of 13. We note that this is simply the calculation of shortest paths according to MIC in the special case where w2 ¼ 1. Furthermore, note that this constraint can be expressed as a regular expression, and is thus solvable using an algorithm for the regular language constrained shortest path problem. It is easiest to express this constraint in terms

C.B. Ward, N.M. Wiegand / Computer Networks 54 (2010) 208–217

of the equivalent finite state machine. Simply, each state in the finite state machine represents the last wireless channel used in the path. From this state, it may transition to every other state with the next edge, but may not remain in this state (reuse the wireless channel on the next hop). Now, although this particular example only considers the equivalence of shortest MIC paths when w2 ¼ 1, we will now demonstrate that the algorithm which solves the regular-language constrained path problem (developed independently by Barrett et al. [1] and Yannakakis [3]) can be easily modified to work with arbitrary w2 values. This canonical algorithm works by taking the cross product of the input graph and the graph representing the finite state machine of the regular expression given as input. In the cross product graph, a vertex represents the combination of both a vertex and an FSM state. Edges in the cross product graph correspond to crossing an edge in the original graph which results in a valid transition in the FSM. Consider a graph G with two vertices v1 and v2 and a finite state machine M with two states s1 and s2 . Then, the cross product G  M of G and M will be a graph with four states: ðv1 ; s1 Þ; ðv1 ; s2 Þ; ðv2 ; s1 Þ, and ðv2 ; s2 Þ. G  M will have an edge ððvi ; sj Þ; ðvk ; sm ÞÞ if and only if there exists an edge ðvi ; vk Þ in G with some label ‘ and there exists a transition ðsj ; sm Þ on label ‘ in M. In the specific case of this problem, the states of the machine would correspond to the possible values for the last wireless channel used in the current path. Thus, an edge ððvi ; sj Þ; ðvk ; sm ÞÞ would exist only if there is an edge ðvi ; vk Þ labeled with the wireless channel corresponding to state sm and sj – sm . Once the cross-product graph is obtained, the shortest path can be determined by running a traditional shortest path algorithm on the cross-product graph. That is, suppose that we wish to find the shortest regular language constrained shortest path from vertex u to vertex v. Then, this is simply the shortest path in the cross product graph from ðu; sÞ to ðv; f Þ where s is the start state and f is any final state. Hence, we can directly reduce the computation of shortest MIC paths to the regular language shortest path problem in the case of w2 ¼ 1. In order to handle the case when w2 – 1, consider that the lack of edges corresponding to violations of the finite state machine correspond to infinite w2 weights. By adding edges in the cross product which correspond to a violation of the state machine’s input language, we can ensure that these edges have the appropriate (non-infinite) w2 weight. That is, instead of disallowing edges which correspond to valid edges in the original graph but violate the regular language, we instead allow but raise the cost of the edge by w2 . In this case, the resulting graph is very similar to the virtual node construction given for the polynomial time algorithm given by Yang, et al. Furthermore, we observe that any set of forbidden adjacent labels can be expressed in a finite state machine of size at most OðjRj2 Þ (jRj states and OðjRj2 Þ transitions), and so the slight generalization of the problem which will be described in Section 5 can be solved in a similar fashion when considering shortest (not simple) paths. Moreover, it can be solved without introducing additional computational complexity. This analogy to an existing set of problems is obviously helpful, as it gives us a set of existing polynomial time

213

algorithms which can be easily modified to compute shortest paths over routing metrics with channel assignment constraints. From this, we can also see that a number of potentially useful modifications are computable by slight modification of this algorithm. For example, we can trivially extend the ‘‘memory” of channel history, at the expense of creating a larger finite state machine. We can also express an arbitrary set of conflicting wireless channels, each pair with its own penalty value. This could allow a metric capable of expressing interference caused by partially overlapping channels, such as could occur in a 802.11b/g network. Additionally, some of the results of formal language constrained path problems can be applied to computing routing metrics of this type. For a simple example, as noted originally by Mendelzon and Wood [2], the regular-language-constrained simple path problem is solvable in polynomial time on acyclic graphs, and hence this is also true for both MIC and our generalization. As another, Barrett et al. [13] indicates that for certain subsets of regular languages (linear regular languages), the efficiency of the regular-language-constrained shortest path algorithm may be improved by not calculating the entire cross-product graph in advance, and merely computing pieces of the graph as required [13]. Although not identical, the regularlanguage corresponding to MIC is similar in construction to a linear regular language, and it is likely that this same increase in efficiency could be achieved. Finally, our discussion in Sections 4 and 5 is heavily influenced by the terminology and results of labeled path problems. From this basis, we will now discuss our complexity results. 4. Most used label shortest paths In examining the WCETT metric, we note that the first term can be simply computed, as it is nothing more than a simple weighted shortest path problem. It is the second component of the metric, which seeks to minimize the most used channel, which provides difficulties. It can be verified that the addition of the first component does not fundamentally alter the problem of computing a shortest path over the metric, and so we consider the following labeled shortest path decision problem corresponding to the second term of the WCETT metric: Definition 6. The most used label shortest paths (MUL) decision problem: Input: Directed Multigraph G ¼ ðV; EÞ, a set of labels R, weight function U : E ! R , labeling L : E ! R, source s 2 V, destination t 2 V, maximum weight W. Output: True if 9 path psuch that mulðpÞ 6 W where

mulðpÞ ¼ maxfX j ðpÞg; j2C X :UðeÞ X j ðpÞ ¼ e2p labelðeÞ¼j

False, otherwise.

214

C.B. Ward, N.M. Wiegand / Computer Networks 54 (2010) 208–217

Theorem 7. The MUL Problem is NP-Complete. Proof. (: We first note that we can verify in polynomial time whether for a given path p it is the case that mulðpÞ < W. Thus the problem is in NP. ): To show that the MUL problem is NP-Hard we reduce from K-Bin Packing [19]. Suppose we have an instance of the K-Bin Packing Problem in which we have k unit bins fB1 ; . . . ; Bk g and n items f1; . . . ; ngwith weights fI1 ; . . . ; In g. The K-Bin Packing Problem asks whether it is possible to assign items to bins such that total weight in each bin is 6 1. From this input we will construct the input to an instance of the MUL problem. Let R ¼ f1; . . . ; kg, and create the graph G ¼ ðV; EÞ. Here V ¼ fv1 ; . . . ; vnþ1 g. For each vertex vi ð1 6 i 6 nÞ we create edges ei;j ¼ ðvi ; viþ1 Þ ð1 6 j 6 kÞ in E. Then let Lðvi;j Þ ¼ j and Uðvi;j Þ ¼ Ii . Finally, set s ¼ v1 ; t ¼ vnþ1 , and W ¼ 1. As Fig. 2 shows, the reduction graph is essentially a linear chain from source to sink where the choice of label (wireless channel) in each hop corresponds to choosing a bin for each element in the original bin packing problem. The minimization of the most used wireless channel thus yields the best bin packing. To formally complete the reduction, we run MUL on ðG; R; U; L; s; t; WÞ. If MUL returns true, then the shortest path through G represents a bin-packing, as each element has a label representing the bin it is in, and mulðpÞ is the total weight of all elements put in the most filled bin. h Fact 8. Antenna-interference even on non-overlapping channels implies that practical mesh networks should have a restricted number of interfaces per node. However, bin packing remains NP-Complete even with only two bins [19]. This implies that computing WCETT is an NP-Complete optimization problem even with only 2 wireless radios per node and only 2 wireless channels. By Lemma 3, of course, modified Dijkstra is still a jCj-approximator when jCj ¼ 2. Corollary 9. The MUL Problem is Strong NP-Complete over general undirected graphs, directed acyclic graphs, and series parallel graphs. The MUL Problem is Strong NP-Complete for any fixed number of labels k P 2.

Proof. Note that if any shortest mul-path contains a loop, we can remove the loop and it remains a shortest mulpath. Thus, the reduction in Theorem 7 holds if the edges are undirected. Furthermore, the graph given in Theorem 7 is both a directed acyclic graph and a series parallel graph, and thus the problem is NP-Hard over these sets of input graphs as well. K-Bin packing is Strong NP-Hard for k P 2, and our reduction is a pseudo-polynomial transformation according to the four necessary properties defined by Garey and Johnson [19]. These four conditions are that:  The reduction is a mapping reduction.  The reduction takes time polynomial in terms of the size of the original problem.  The size of the resulting problem is within a polynomial bound of the original problem.  All weights in the resulting problem are within a polynomial bound of the original problem. As all these conditions hold, MUL is Strong NP-Hard for any number of labels P 2 over all the aforementioned graph classes. h Fact 10. Recall that NP-Completeness in the strong sense implies that there exists no pseudo-polynomial time algorithm for the problem [19]. Thus, there exists no pseudopolynomial time algorithm for computing the shortest WCETT path on general directed, general undirected, directed acyclic, or series parallel graphs for any number of wireless channels k P 2.

5. Restricted adjacent-labels simple paths The MIC metric attempts to minimize intra-flow interference by increasing the cost of using the same wireless channel on two consecutive network links in a path. However, this interference is not necessarily restricted to transmissions on the same channel. Take, for example, 802.11g wireless networks where each of the 14 channels (11 in the US, due to legal restrictions) causes interference to between 4 and 8 other channels [18]. Thus, we generalize this to the case in which each channel may interfere with some number of other channels, and thus increase the cost of a transmission which uses consecutive edges labeled with conflicting channels.

Fig. 2. Reduction from the K-bin packing problem.

C.B. Ward, N.M. Wiegand / Computer Networks 54 (2010) 208–217

The following formally expresses this generalization: Definition 11. The restricted adjacent-labels simple path decision problem (RASP): Input: Directed or Undirected Multigraph G ¼ ðV; EÞ, a set of labels R, weight function U : E ! R , weights w1 ; w2 2 Rþ , labeling L : E ! R, k unordered pairs of labels F ¼ fða1 ; b1 Þ; . . . ; ðak ; bk Þg where ai 2 R and bi 2 R for all 1 6 i 6 k, source s 2 V, destination t 2 V,maximum weight W. Also define CSC as in Section 2.2. ; if e begins p. Output: True if 9 path p such that ralðpÞ 6 W where

ralðpÞ ¼

X ðUðeÞ þ CSC e;p Þ e2p

False, otherwise. Note that in the case where the set of restricted pairs is fðl1 ; l1 Þ; . . . ; ðljRj ; ljRj Þg, each li 2 R, we have exactly the MIC metric given in Yang et al. [7]. Corollary 14 shows that this is NP-Complete for directed graphs. Also note that, as discussed earlier, in the case of shortest (rather than simple) paths, this problem is solvable in polynomial time, as given in [7]. However, this result implies the impossibility of improving upon this flaw in the MIC metric and thus emphasizes the importance of avoiding the problem in practice by proper choice of w1 and w2 values. In order to show that RASP is NP-Complete, we reduce from the Shortest Paths with Forbidden Pairs Problem (SPFP). SPFP was shown to be NP-Complete by Gabow et al. [20]. The input to the SPFP Problem consists of a graph and a set of pairs of vertices. The output of the problem is the shortest path in an input graph subject to the constraint that at most one vertex from each pair may appear in the path. Formally, we define the following decision version of SPFP below: Definition 12. Shortest paths with forbidden pairs:

stitution of two directed edges for each undirected edge). The DSPFP Problem takes as input a graph G with weight function U, start vertex s, destination vertex t; k pairs of vertices fða1 ; b1 Þ; ða2 ; b2 Þ; . . . ; ðak ; bk Þg, and maximum weight W. The intuition of the reduction is as follows. We wish to modify the input graph such that the adjacent labeling constraints will enforce the forbidden pairs constraint for us. In order to do this, we will replace each vertex with an input and an output vertex connected through a chain of conflict vertices. Each of these conflict vertices corresponds to a forbidden pair in the input problem which this vertex is a member of. That is, if the vertex is part of three conflicts, it will have three intermediate conflict vertices between its input and output. These conflict vertices will be shared with the other vertices in the forbidden pairs. Fig. 3 shows two vertices, x and y, which are each part of a single forbidden pair with each other. The labels given to these edges, and the set of forbidden adjacent labels, will be set such that it is impossible for a path to ‘‘jump” to another part of the graph by way of the conflict vertices. The result of these shared conflict vertices is that, as the problem requires the solution to be a simple path, each conflict vertex can be used only once in the solution. Hence, the simple solution path, when translated back to the original graph, will not violate the input set of forbidden pairs. A more formal description of the reduction follows. We will construct graph H ¼ ðV 0 ; E0 Þ for input to the RASP Problem. Begin by setting V 0 ¼ ;; E0 ¼ ;; F ¼ ;. Also, let us denote the vertices of V as f1; . . . ; jVjg, and let R ¼ f1; . . . ; 2jVj; x; wg. Here, x and w are additional labels we will need in the graph construction. For each vertex v 2 V add three vertices vS ; vI , and vO to V 0 . For each conflict pair ðaj ; bj Þ add vertex C j to V 0 . For each vertex v 2 V which is in c conflict pairs, add c  1 vertices vC j ð1 6 j < c). For each edge ðu; vÞ 2 E add edge ðuO ; vI Þ to E0 ; let LðuO ; vI Þ ¼ x. For each vertex v 2 V, add edge ðvS ; vI Þ to E0 ;

Input: Directed Graph G ¼ ðV; EÞ, weight function U : E ! Rþ , pairs of vertices fða1 ; b1 Þ; . . . ; ðak ; bk Þg, maximum weight W. P Output: True if 9 path p such that e2p UðeÞ < W and 8 vertices u; v in path p there is no k such that u  ak and v  bk .

Theorem 13. The RASP Problem is NP-Complete for general directed and undirected graphs. Proof. (: We first note that we can verify whether for a given path p it is the case that ralðpÞ < W. Thus, the problem is in NP. ): We reduce from the Directed Shortest Paths with Forbidden Pairs Problem [20] to the RASP Problem with an undirected input graph (as the undirected case reduces directly to the directed case by the sub-

215

Fig. 3. Conflicting vertices in reduction from the DSPFP.

216

C.B. Ward, N.M. Wiegand / Computer Networks 54 (2010) 208–217

let LðvS ; vI Þ ¼ x. If for each vertex v 2 V, v is not in any conflict pair, add edge ðvI ; vO Þ to E0 ; let LðvI ; vO Þ ¼ w. For all vertices v 2 V, let m be the number of conflict pairs for v, and add edges fðvI ; C 1 Þ; ðC 1 ; vC 1 Þ; ðvC 1 ; C 2 Þ; . . . ; ðC m ; vO Þg to E0 . Let LðvI ; C 1 Þ ¼ LðvC 1 ; C 2 Þ ¼    ¼ LðvC m1 Þ ¼ v and in each conflict pair i, let v be ak , then set LðC i ; vC i Þ ¼ jVj þ bk , jVj plus the label corresponding to the other vertex in this conflict pair. Let the weight function U0 on E0 be such that for u; v 2 V edges ðuO ; vI Þ have weight from the instance of the DSPFP problem, and all other edges have weight 0. Finally, for all vertices v 2 V add ðv; jVj þ vÞ to F, and for each u which conflicts with v add ðv; uÞ. Also add ðx; xÞ to F, and set w1 ¼ 0 and w2 ¼ 1. Now, return the solution from calling RASP with input ðG0 ; U0 ; w2 ; L; F; sS ; tO ; WÞ. Fig. 3 shows the construction of a pair of vertices x and y which conflict only with each other. There will exist a simple path with weight 6 W from sI to t O in G0 which does not pass over any consecutive labels listed in F if and only if it is possible to find a path from s to t in G with weight 6 W. This follows from the fact that if two vertices u and v conflict, any path which uses both u and v must pass through the conflict vertex corresponding to their conflict twice, violating simplicity of the resultant path. This follows from the fact that the x labeling prevents the solution from creating a path which has no analogue in G. This is because all edges with analogue in G have label x, forcing a path to pass through all conflict vertices associated with a vertex. Thus, the shortest simple path from sS to tO in G0 will correspond exactly to the shortest path from s to t in G in which no forbidden pairs of vertices are used, and these paths will have the same weight. h Corollary 14. The RASP Problem is NP-Complete for general directed graphs and the given set of forbidden adjacent labels fðl1 ; l1 Þ; . . . ; ðljRj ; ljRj Þg (li 2 R). Proof. In this case, we note that our previous reduction holds but that the four edges surrounding conflict vertices can no longer be appropriately constrained using labels. This is because there are two undirected edges which could allow a ‘‘leak” of the path through the conflict vertices, and the set of forbidden adjacent labels allows us to plug one of these at a time. In this case, the directedness supplies the necessary constraint, demonstrating that the problem remains NP-Complete in this subcase on general directed graphs. h

5.1. Approximation of RASP In this section, we demonstrate that RASP belongs to a class of NP-Complete optimization problems (NPO PBComplete) which are only poorly approximable. This implies that MIC can have arbitrarily bad worst case behaviour when compared to the optimal simple path if weights are not chosen appropriately to mitigate the problem. The complexity class NPO PB consists of NP optimization problems in which the objective function being optimized is bounded by some polynomial in terms of the

size of the problem input [21]. The complete problems of the NPO PB class are those which every NPO PB problem can be reduced to in polynomial time using an approximation preserving reduction. An approximation preserving reduction is simply a reduction wherein an approximation of the problem reduced to yields an approximation for the original problem. Thus, any hardness of approximation results from the original problem can be applied. The complete problems of this class were formerly divided into Min PB-Complete and Max PB-Complete, corresponding to the set of complete minimization and complete maximization problems, respectively. However, it has been shown that these are equivalent, and so the term NPO PB-Complete is used [22]. The NPO PB-Complete problems are difficult to approximate. This can be observed by a result of Halldórsson, which showed that the calculating the Minimum Maximal Independence Number is not approximable within a factor of n1 for any  > 0 [23]. As this problem belongs to the NPO PB-Complete class, and all NPO PB-Complete problems are approximation preserving polynomially reducible to each other, this result implies that NPO PB-Complete problems are difficult to approximate [21]. Theorem 15. The optimization version of the RASP Problem is NPO PB-Complete for general directed graphs even with the given set of restricted labels fðl1 ; l1 Þ; . . . ; ðljRj ; ljRj Þgðli 2 RÞ, provided that the weights w1 ; w2 , and the weight function w are restricted to some polynomial of the input size. Proof. The Shortest Paths with Forbidden Pairs Problem is NPO PB-Complete by the work of Kann [21]. Thus, in order to show that the RASP Problem is NPO PB-Complete it suffices to show that the reduction given in Theorem 13 and modified in Corollary 14 is approximation preserving, and that the objective function of the RASP optimization problem is bounded by some polynomial function of the input size. (: First note that the original reduction preserves the cost of a path, as the edges added to enforce the pair restriction do not contribute additional cost to the path. Any valid path computed between start and end vertices in the RASP graph will correspond exactly to a valid path in the original SPFP graph, and have the same cost. Hence, any approximate path in the RASP graph will correspond to an approximate path in the SPFP problem, and have exactly the same approximation ratio. ): Since we consider only simple paths, it is straightforward to observe that the objective function is bounded by at most n  1 times the largest edge weight in the graph. Provided that the input weights are restricted to some polynomial function of the input size, the objective function is thus, also restricted to a polynomial function of the input size. h Since NPO PB-Complete problems cannot be approximated within n1 for any  > 0, then RASP is difficult to approximate. As a result, we can observe that on directed

C.B. Ward, N.M. Wiegand / Computer Networks 54 (2010) 208–217

graphs, simple MIC paths are, likewise, difficult to approximate. This implies that the work of Yang et al., in which shortest MIC paths are computed, is the theoretical limit of this particular approach to metric design, within the bounds of polynomial-time computability. 6. Conclusions In this paper we have demonstrated the NP-Completeness of calculating shortest loop-free routing paths over WCETT and MIC, which yield good routing solutions in multichannel wireless networks. We have shown that computing shortest WCETT paths is strongly NP-Complete, and thus there exists no pseudo-polynomial time algorithm, but we have also discussed the provable approximability of the modified Dijkstra’s algorithm, as given by Draves et al. [8]. Additionally, we have demonstrated that computing simple MIC paths belongs to a class of optimization problems which are difficult to approximate. This implies that good choice of weights is important to avoid this theoretical bad behaviour. Furthermore, we have also related the calculation of these metrics to a class of existing problems, the formal language constrained path problems, and noted that an entire class of similar wireless routing metrics have polynomial-time algorithms simply derivable from algorithms from this class. We believe that there is much to be gained by viewing multi-radio multi-hop wireless metrics through the lens of labeled graph theory.

[11]

[12] [13]

[14] [15]

[16]

[17]

[18]

[19] [20]

[21] [22]

[23]

217

Wireless Communications, Networking and Mobile Computing, 2007, pp. 1550–1553. Wei Zhou, Dongbo Zhang, Daji Qiao, Comparative study of routing metrics for multi-radio multi-channel wireless networks, in: Wireless Communications and Networking Conference, 2006, WCNC 2006, vol. 1, IEEE, 2006, pp. 270–275. Serge Abiteboul, Victor Vianu, Regular Path Queries with Constraints, 1997, pp. 122–133. Chris Barrett, Keith Bisset, Riko Jacob, Goran Konejevod, Madhav Marathe, Classical and contemporary shortest path problems in road networks: implementation and experimental analysis of the transims router, in: ESA 2002, LNCS 2461, 2002, pp. 126–138. Mike Beaven, Ryan Stansifer, Explaining type errors in polymorphic languages, ACM Lett. Program. Lang. Syst. 2 (1–4) (1993) 17–30. Bruce McAdam, On the unification of substitutions in type inference, in: Implementation of Functional Languages: 10th International Workshop IFL’98, Springer, Berlin/Heidelberg, 1999, pp. 649–650. Phillip G. Bradford, Quickest path distances on context-free labeled graphs, in: Sixth WSEAS Conference on Computational Intelligence, Man-Machine Systems and Cybernetics, Information, Security and Privacy (ISP ’07), 2007, pp. 22–29. M.H. Moore, On the fastest route for convoy-type traffic in flowrateconstrained networks, Transportation Science 10 (2) (1976) 113– 124. IEEE 802.11 Working Group (2007-06-12). IEEE 802.11-2007: Wireless LAN Medium Access Control (MAC) and Physical Layer (PHY) Specifications. Michael Garey, David Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness, W.H. Freeman, 1979. Harold Gabow, Shachindra Maheshwari, Leon Osterweil, On two problems in the generation of program test paths, IEEE Trans. Softw. Eng. 2 (3) (1976) 227–231. Viggo Kann, Polynomially bounded minimization problems that are hard to approximate, Nordic J. Comput. 1 (3) (1994) 317–331. Viggo Kann, On the approximability of NP-complete optimization problems, PhD Thesis, Royal Institute of Technology, Stockholm, Sweden, 1992. Magnús M. Halldórsson, Approximating the minimum maximal independence number, Inf. Process. Lett. 46 (4) (1993) 169–172.

Acknowledgement Special thanks to Phillip G. Bradford, for his numerous suggestions which have improved this paper enormously. References [1] Chris Barrett, Riko Jacob, Madhav Marathe, Formal language constrained path problems, SIAM Journal on Computing 30 (3) (2001) 809–837. [2] Alberto Mendelzon, Peter Wood, Finding regular simple paths in graph databases, SIAM J. Comput. 24 (6) (1995) 1235–1258. [3] Mihalis Yannakakis, Graph-theoretic methods in database theory, in: PODS’90: Proceedings of the Ninth ACM SIGACT-SIGMOD-SIGART Symposium on Principles of Database Systems, ACM Press, New York, NY, USA, 1990, pp. 230–242. [4] Mihalis Yannakakis, Perspectives on database theory, SIGACT News 27 (3) (1996) 25–49. [5] Phillip G. Bradford, David A. Thomas, Labeled shortest paths in digraphs with negative and positive edge weights, RAIRO-Theor. Inf. Appl. 43 (3) (2009) 567–583. [6] Venkatesh Choppella, Christopher T. Haynes, Source-tracking unification, Inf. Comput. 201 (2) (2005) 121–159. [7] Yaling Yang, Jun Wang, Robin Kravets, Designing routing metrics for mesh networks, in: IEEE Workshop on Wireless Mesh Networks, WiMesh, 2005. [8] Richard Draves, Jitendra Padhye, Brian Zill, Routing in multi-radio, multi-hop wireless mesh networks, in: MobiCom’04: Proceedings of the 10th Annual International Conference on Mobile Computing and Networking, ACM Press, New York, NY, USA, 2004, pp. 114–128. [9] Douglas S.J. De Couto, Daniel Aguayo, John Bicket, Robert Morris, A high-throughput path metric for multi-hop wireless routing, in: Proceedings of the 9th ACM International Conference on Mobile Computing and Networking (MobiCom’03), San Diego, California, September 2003. [10] Weirong Jiang, Shuping Liu, Yun Zhu, Shiming Zhang, Optimizing routing metrics for large-scale multi-radio mesh networks, in:

Charles Ward received his B.S. in Mathematics and Ph.D. in Computer Science from the University of Alabama in 2004 and 2008, respectively. He is currently a postdoctoral associate in Dr. Steven Skienas lab at Stony Brook University. His research areas include algorithms, graph theory, networking, formal languages, computer security, and distributed algorithms.

Nathan Wiegand received his B.S. in Mathematics and M.S. in Computer Science from the University of Alabama in 2004 and 2007, respectively. He is currently a Ph.D. candidate in the Department of Computer Science at the University of Alabama. His research areas include algorithms, graph theory, programming languages, networking, formal languages, distributed algorithms, and approximability.