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Finding Cycles and Trees in Sublinear Time Oded Goldreich Weizmann Institute of Science Joint work with Artur Czumaj, Dana Ron, C. Seshadhri, Asaf Shapira, and Christian Sohler

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Preliminaries (boring, but needed…) Consider algorithms in the bounded-degree graph model (for a fixed degree bound, d): The algorithms use queries of the form (v,i) (where i d) that are answered with the i th neighbor of v. Distances are measured as fractions of the maximum possible number of edges (i.e., dN/2). For simplicity, far = being (1)-far. (The results extend to the case that the algorithm is given a proximity parameter, but then the complexity depends on.) cycle = simple cycle

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Our results at a glance – take 1 (naïve) THM 1: An Õ(N 1/2 )-time algorithm for finding (small) cycles in N-vertex graphs that are far from being cycle-free. THM 2: For every fixed k>3, an Õ(N 1/2 )-time algorithm for finding (small) cycles of length at least k in N-vertex graphs that are far from having no cycles of such length. Optimality: No o(N 1/2 )-query algorithm can find such cycles. THM 3: For every fixed k>1, an O(1)-time algorithm for finding trees with at least k leaves in graphs that are far from having no such trees. In the bounded-degree graph model cycle = simple cycle small = polylog(N)-size

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Our results at a glance – take 2 (minors) THM 1: An Õ(N 1/2 )-time algorithm for finding (small) C 3 -minors in N-vertex graphs that are far from being C 3 -minor free. THM 2: For every fixed k>3, an Õ(N 1/2 )-time algorithm for finding (small) C k -minors in N-vertex graphs that are far from being C k -minor free. Optimality: For any H that contains a cycle, no o(N 1/2 )-query algorithm can find H-minors in a N-vertex graphs that is far from being H-minor free. THM 3: For every fixed k>1, an O(1)-time algorithm for finding T k -minors in graphs that are far from T k – minor free, where T k denotes the k-vertex star. THM 4: For any cycle-free H, an O(1)-time algorithm for finding H-minors in graphs that are far from H – minor free. In the bounded-degree graph model cycle = simple cycle C k = k-vertex cycle Def: A graph G has an H-minor if H can be obtained from G by vertex and edge removal and edge contraction. Dichotomy: H with/w.o. cycles.

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One-sided error testing and finding structures Observation: When a one-sided error tester for a property rejects a graph G, it must be the case that the subgraph viewed by the tester is inconsistent with any graph in. In some cases, this subgraph has a natural appeal. E.g., if is being bipartite, then the subgraph must be a non-bipartite graph; if is being H-minor free, then the subgraph must be an H-minor. Thus, all our results can be stated in terms of results regarding one-sided error testers (see next slide…). Recall that two-sided error testers of O(1)-time are known for H-minor freeness (cf. [BSS] vastly extending [GR] ). NTS, these do not yield algorithms for finding minors. Dichotomy: one/two-sided error.

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Our results at a glance – take 3 (1-sided-error-testers) THM 1: An Õ(N 1/2 )-time one-sided error tester for C 3 -minor freeness (a.k.a cycle-freeness). THM 2: For every fixed k>3, an Õ(N 1/2 )-time one-sided error tester for C k -minor freeness. Optimality: For any H that contains a cycle, no o(N 1/2 )-query one-sided error tester for H-minor freeness. Yet, an O(1)-time two-sided error tester exists (cf. [BSS])! THM 4: For any cycle-free H, an O(1)-time one-sided error tester for H – minor freeness. In the bounded-degree graph model cycle = simple cycle C k = k-vertex cycle Def: A graph G has an H-minor if H can be obtained from G by vertex and edge removal and edge contraction.

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Techniques: testing cycle-freeness THM 1: An Õ(N 1/2 )-time one-sided error tester for cycle-freeness. In the bounded-degree graph model cycle = simple cycle Idea: randomly reduce testing cycle-freeness to testing bipartiteness, by replacing each edge with a 2-path w.p. ½ (and leaving it intact otherwise). A cycle-free graph is always mapped to a bipartite graph, whereas each cycle is mapped with probability ½ to an odd cycle. CLM: A graph that is -far from being cycle-free is mapped, w.v.h.p, to a graph that is ( )-far from being bipartite. Detail: local implementation of the reduction. Operations of the bipartite-tester are emulated via queries to the original graph. The two-sided error tester just compares the # of edges to the # of cc.

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Techniques: testing C k -minor freeness THM 2: An Õ(N 1/2 )-time one-sided error tester for C k -minor freeness. In the bounded-degree graph model cycle = simple cycle C k = k-vertex cycle Idea: (deterministically) reduce testing C k -minor freeness to testing cycle-freeness, by replacing shorter than k with adequate gadgets. E.g., for k=4, replace each triangle by a 3-star. A C 4 -minor free graph is always mapped to a cycle-free graph, whereas any C 4 -minor is mapped to a cycle. CLM: A graph that is -far from being C 4 -minor free is mapped to a graph that is ( )-far from being cycle-free. Detail: local implementation of the reduction. Operations of the bipartite-tester are emulated via queries to the original graph. Two triangles sharing an edge contain a cycle of length four. A C 4 -minor free graph is a tree of triangles and edges. (See next slide)

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Testing C k -minor freeness: auxiliary figures Two triangles sharing an edge contain a cycle of length four. They are replaced by edges that contain a 4-cycle. A C 4 -minor free graph is a tree of triangles and edges. These triangles disappear (in repl t) and the tree remains. The reduction replaces red edges by blue edges. (Black edges remain intact.)

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Techniques: testing T k -minor freeness THM 3: An poly(k/ )-time one-sided error tester for T k -minor freeness. In the bounded-degree graph model T k = the k-vertex star The tester performs a BFS from a randomly chosen start vertex till either encountering k vertices in a layer or visiting 4k/ layers. Accept iff the explored subgraph is T k -minor free. Call a vertex v bad if it is contained in a set S such that the subgraph induced by S contains a T k -minor and has radius at most 4k/ from v. Observe that if the graph has few bad vertices, then it is close to being T k -minor free (by isolating all bad vertices and omitting the edges that separate each 4k/ -depth BFS from the rest of the graph).

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The End The slides of this talk are available at The paper itself is available at

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