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Certifying algorithms Algorithms and Networks

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Certifying algorithms What is it? Examples: –Euler tour –Bipartite graphs –Flow –Planarity Certifying algorithms2

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Type of algorithm design, introduced by Mehlhorn and others No formal definition here Certifying algorithms3

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Correctness of programs Program: –Input –(Working of program): Hidden to user –Output But... how do we know there is no bug? Why is the program correct? Certifying algorithms4

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Example Euler tour algorithm Input: Graph G=(V,E) Output: –An Euler tour if it exists –“NO”, if there is no Euler tour Certifying algorithms5

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Example Euler tour algorithm Input: Graph G=(V,E) Output: –An Euler tour if it exists This, we can check! –“NO”, if there is no Euler tour But how do we know if this answer is correct? Certifying algorithms6

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Certifying algorithm Program –Input –(Working of program): still hidden to user –Output + proof (certificate) that output is correct! Plus: easy way to check these proofs Certifying algorithms7

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Euler tour Certificate for Euler tour program YES: the Euler tour itseld NO: … Certifying algorithms8

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Euler theorem A graph G=(V,E) has an Euler tour, if and only if the following two conditions hold: –Every vertex has even degree –G is connected Certifying algorithms9

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No-Certificate for Euler tour A vertex of odd degree Or A set of vertices W that forms a connected component –Verify: W has less than n vertices, and all neighbors of all vertices in W belong to W Certifying algorithms10

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Bipartite graphs A graph G=(V,E) is bipartite if we can partition V into sets X and Y such that each edge in E has one endpoint in X and one endpoint in Y So … what would be a certificate for Bipartiteness? Certifying algorithms11

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Bipartite certificate Yes: bipartition No: odd cycle in G Certifying algorithms12

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Algorithm Make a depth first search spanning forest of G Color the vertices on odd levels red and on even levels blue Is there an edge between two vertices of the same color? Then: take the path between these in the tree, and add the edge: odd cycle Certifying algorithms13

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Flow Input: flow network (G, s, t, c) Output: maximum flow f So, what is a certificate? Certifying algorithms14

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Certificate for flow Pair: –The flow function f –And, the corresponding maximum cut See (the proof of) the max flow min cut theorem Certifying algorithms15

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Bipartite Matching Input: Bipartite graph G=(V U W,E) Output: A maximum matching in G What is a certificate for the matching? Certifying algorithms16

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Certificate for bipartite matching Look at the flow model from bipartite matching, and a minimum cut. You get: A vertex cover C with the property that |C|=|M| –I.e., each edge has one endpoint in C Details on the board Certifying algorithms17

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Planarity Planar graphs: can be drawn on the plane without crossing edges Many applications! Complicated linear time algorithms for checking planarity –Tarjan, 1976: linear time for checking planarity –Chiba et al, 1985: linear time algorithm that finds a planar embedding Certifying algorithms18

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Kuratowski Theorem (Kuratowski) A graph is planar, if and only if it does not contain a subgraph that is a subdivision of K 5 or K 3,3 Certifying algorithms19

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Subdivision Add vertices on edges Certifying algorithms20

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Wagner Theorem (Wagner) A graph is planar, if and only if it does not contain K 5 or K 3,3 as a minor Certifying algorithms21

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Certificate for nonplanarity Hundack et al. 1987: linear time algorithm that finds a a subgraph that is a subdivision of K 5 or K 3,3 of a nonplanar graph Certifying algorithms22

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Certificate for planarity Schnyder 1990: –Finds in O(n) time an embedding of a graph in the plane, such that Each vertex has integer coordinates between 1 and n – 2 Each edge is a straight line One can easily verify that such an embedding is indeed correct. Certifying algorithms23

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Certifying algorithm for planarity Either output Schnyder embedding OR Output minor Certifying algorithms24

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Conclusions Certifying algorithms Useful for reliability of software Some examples … Implemented in some packages nowadays, e.g., LEDA Certifying algorithms25

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What is the next line of the proof? a). Let G be a graph with k vertices. b). Assume the theorem holds for all graphs with k+1 vertices. c). Let G be a.

What is the next line of the proof? a). Let G be a graph with k vertices. b). Assume the theorem holds for all graphs with k+1 vertices. c). Let G be a.

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