Uri Zwick

Color-coding

We describe a novel randomized method, the method of color-coding for finding simple paths and cycles of a specified length k, and other small subgraphs, within a given graph G = (V,E). The randomized algorithms obtained using this method can be derandomized using families of perfect hash functions. Using the color-coding method we obtain, in particular, the following new results: • For every fixed k, if a graph G = (V,E) contains a simple cycle of size exactly k, then such a cycle can be found in either O(V ) expected time or O(V ω log V ) worst-case time, where ω < 2.376 is the exponent of matrix multiplication. (Here and in what follows we use V and E instead of |V | and |E| whenever no confusion may arise.) • For every fixed k, if a planar graph G = (V,E) contains a simple cycle of size exactly k, then such a cycle can be found in either O(V ) expected time or O(V log V ) worst-case time. The same algorithm applies, in fact, not only to planar graphs, but to any minor closed family of graphs which is not the family of all graphs. • If a graph G = (V,E) contains a subgraph isomorphic to a bounded tree-width graph H = (VH , EH) where |VH | = O(log V ), then such a copy of H can be found in polynomial time. This was not previously known even if H were just a path of length O(log V ). These results improve upon previous results of many authors. The third result resolves in the affirmative a conjecture of Papadimitriou and Yannakakis that the LOG PATH problem is in P. We can show that it is even in NC.

Compact routing schemes

We describe several compact routing schemes for general weighted undirected networks. Our schemes are simple and easy to implement. The routing tables stored at the nodes of the network are all very small. The headers attached to the routed messages, including the name of the destination, are extremely short. The routing decision at each node takes <i>constant</i> time. Yet, the <i>stretch</i> of these routing schemes, i.e., the worst ratio between the cost of the path on which a packet is routed and the cost of the cheapest path from source to destination, is a small constant. Our schemes achieve a near-optimal tradeoff between the size of the routing tables used and the resulting stretch. More specifically, we obtain:<ul><li>A routing scheme that uses only O (<i>n</i> ^1/2) bits of memory at each node of an <i>n</i>-node network that has stretch 3. The space is <i>optimal</i>, up to logarithmic factors, in the sense that every routing scheme with stretch < 3 must use, on some networks, routing tables of total size &OHgr;(<i>n</i>²), and every routing scheme with stretch < 5 must use, on some networks, routing tables of total size &OHgr;(<i>n</i>^3/2). The headers used are only (1 + <i>&Ogr;</i>(1)) log²> <i>n</i>-bits long and each routing decision takes <i>constant</i> time. A variant of this scheme with [log<subscrpt>2</subscrpt> <i>n</i>] -bit headers makes routing decisions in <i>&Ogr;</i>(log log <i>n</i>) time. </li><li>Also, for every integer <i>k</i> > 2, a general <i>handshaking</i> based routing scheme that uses O (<i>n</i>^1/k) bits of memory at each node that has stretch 2<i>k</i> - 1. A conjecture of Erdös from 1963, settled for <i>k</i> = 3, 5, implies that the routing tables are of near-optimal size relative to the stretch. The handshaking is similar in spirit to a DNS lookup in TCP/IP. Headers are <i>&Ogr;</i>(log² <i>n</i>) bits long and each routing decision takes <i>constant</i> time. Without handshaking, the stretch of the scheme increases to 4<i>k</i> — 5. One ingredient used to obtain the routing schemes mentioned above, may be of independent practical and theoretical interest: </li><li>A shortest path routing scheme for <i>trees</i> of arbitrary degree and diameter that assigns each vertex of an <i>n</i>-node tree a (1 + <i>&Ogr;</i>(1)) log² <i>n</i>-bit label. Given the label of a source node and the label of a destination it is possible to compute, in <i>constant</i> time, the port number of the edge from the source that heads in the direction of the destination. </li></ul> The general scheme for <i>k</i> > 2 also uses a clustering technique introduced recently by the authors. The clusters obtained using this technique induce a sparse and low stretch <i>tree cover</i> of the network. This essentially reduces routing in general networks into routing problems in trees that could be solved using the above technique.

The complexity of mean payoff games on graphs

Abstract We study the complexity of finding the values and optimal strategies of mean payoff games on graphs, a family of perfect information games introduced by Ehrenfeucht and Mycielski and considered by Gurvich, Karzanov and Khachiyan. We describe a pseudo-polynomial-time algorithm for the solution of such games, the decision problem for which is in NP ∩ coNP . Finally, we describe a polynomial reduction from mean payoff games to the simple stochastic games studied by Condon. These games are also known to be in NP ∩ coNP , but no polynomial or pseudo-polynomial-time algorithm is known for them.

Reachability and Distance Queries via 2-Hop Labels

Reachability and distance queries in graphs are fundamental to numerous applications, ranging from geographic navigation systems to Internet routing. Some of these applications involve huge graphs and yet require fast query answering. We propose a new data structure for representing all distances in a graph. The data structure is distributed in the sense that it may be viewed as assigning labels to the vertices, such that a query involving vertices u and v may be answered using only the labels of u and v.Our labels are based on 2-hop covers of the shortest paths, or of all paths, in a graph. For shortest paths, such a cover is a collection S of shortest paths such that for every two vertices u and v, there is a shortest path from u to v that is a concatenation of two paths from S. We describe an efficient algorithm for finding an almost optimal 2-hop cover of a given collection of paths. Our approach is general and can be applied to directed or undirected graphs, exact or approximate shortest paths, or to reachability queries.We study the proposed data structure using a combination of theoretical and experimental means. We implemented our algorithm and checked the size of the resulting data structure on several real-life networks from different application areas. Our experiments show that the total size of the labels is typically not much larger than the network itself, and is usually considerably smaller than an explicit representation of the transitive closure of the network.

All pairs shortest paths using bridging sets and rectangular matrix multiplication

We present two new algorithms for solving the All Pairs Shortest Paths (APSP) problem for weighted directed graphs. Both algorithms use fast matrix multiplication algorithms.The first algorithm solves the APSP problem for weighted directed graphs in which the edge weights are integers of small absolute value in <i>Õ</i>(<i>n</i>^2+μ) time, where μ satisfies the equation ω(1, μ, 1) = 1 + 2μ and ω(1, μ, 1) is the exponent of the multiplication of an <i>n</i> × <i>n</i>^μ matrix by an <i>n</i>^μ × <i>n</i> matrix. Currently, the best available bounds on ω(1, μ, 1), obtained by Coppersmith, imply that μ < 0.575. The running time of our algorithm is therefore <i>O</i>(<i>n</i>^2.575). Our algorithm improves on the <i>&Otilede;</i>(<i>n</i>^(3c+ω)/2) time algorithm, where ω = ω(1, 1, 1) < 2.376 is the usual exponent of matrix multiplication, obtained by Alon et al., whose running time is only known to be <i>O</i>(<i>n</i>^2.688).The second algorithm solves the APSP problem <i>almost</i> exactly for directed graphs with <i>arbitrary</i> nonnegative real weights. The algorithm runs in Õ((<i>n</i>^ω/&epsis;) log(<i>W</i>/&epsis;)) time, where &epsis; > 0 is an error parameter and <i>W</i> is the largest edge weight in the graph, after the edge weights are scaled so that the smallest non-zero edge weight in the graph is 1. It returns estimates of all the distances in the graph with a stretch of at most 1 + &epsis;. Corresponding paths can also be found efficiently.

Fast sparse matrix multiplication

Let <i>A</i> and <i>B</i> two <i>n</i>×<i>n</i> matrices over a ring <i>R</i> (e.g., the reals or the integers) each containing at most <i>m</i> nonzero elements. We present a new algorithm that multiplies <i>A</i> and <i>B</i> using <i>O</i>(<i>m</i>^0.7<i>n</i>^1.2+<i>n</i>^{2+<i>o</i>(1)}) algebraic operations (i.e., multiplications, additions and subtractions) over <i>R</i>. The naïve matrix multiplication algorithm, on the other hand, may need to perform Ω(<i>mn</i>) operations to accomplish the same task. For <i>m</i>≤<i>n</i>^1.14, the new algorithm performs an almost optimal number of only <i>n</i>^{2+<i>o</i>(1)} operations. For <i>m</i>≤<i>n</i>^1.68, the new algorithm is also faster than the best known matrix multiplication algorithm for dense matrices which uses <i>O</i>(<i>n</i>^2.38) algebraic operations. The new algorithm is obtained using a surprisingly straightforward combination of a simple combinatorial idea and existing fast <i>rectangular</i> matrix multiplication algorithms. We also obtain improved algorithms for the multiplication of more than two sparse matrices. As the known fast rectangular matrix multiplication algorithms are far from being practical, our result, at least for now, is only of theoretical value.

Finding and counting given length cycles

We present an assortment of methods for finding and counting simple cycles of a given length in directed and undirected graphs. Most of the bounds obtained depend solely on the number of edges in the graph in question, and not on the number of vertices. The bounds obtained improve upon various previously known results.

A 7/8-approximation algorithm for MAX 3SAT?

We describe a randomized approximation algorithm which takes an instance of MAX 3SAT as input. If the instance-a collection of clauses each of length at most three-is satisfiable, then the expected weight of the assignment found is at least 7/8 of optimal. We provide strong evidence (but not a proof) that the algorithm performs equally well on arbitrary MAX 3SAT instances. Our algorithm uses semidefinite programming and may be seen as a sequel to the MAX CUT algorithm of Goemans and Williamson (1995) and the MAX 2SAT algorithm of Feige and Goemans (1995). Though the algorithm itself is fairly simple, its analysis is quite complicated as it involves the computation of volumes of spherical tetrahedra. Hastad has recently shown that, assuming P/spl ne/NP, no polynomial-time algorithm for MAX 3SAT can achieve a performance ratio exceeding 7/8, even when restricted to satisfiable instances of the problem. Our algorithm is therefore optimal in this sense. We also describe a method of obtaining direct semidefinite relaxations of any constraint satisfaction problem of the form MAX CSP(F), where F is a finite family of Boolean functions. Our relaxations are the strongest possible within a natural class of semidefinite relaxations.

A deterministic subexponential algorithm for solving parity games

The existence of polynomial time algorithms for the solution of parity games is a major open problem. The fastest known algorithms for the problem are randomized algorithms that run in subexponential time. These algorithms are all ultimately based on the randomized subexponential simplex algorithms of Kalai and of Matousek, Sharir and Welzl. Randomness seems to play an essential role in these algorithms. We use a completely different, and elementary, approach to obtain a deterministic subexponential algorithm for the solution of parity games. Our deterministic algorithm is almost as fast as the randomized algorithms mentioned above.

Exact and Approximate Distances in Graphs - A Survey

We survey recent and not so recent results related to the computation of exact and approximate distances, and corresponding shortest, or almost shortest, paths in graphs. We consider many different settings and models and try to identify some remaining open problems.