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Unformatted text preview: Engineering Review Session Jon Turner Computer Science & Engineering Washington University www.arl.wustl.edu/~jst Engineering Big Picture Minimum Fibonacci Shortest spanning trees heaps general greedy method, Prim's algorithm, dheaps how operations are implemented, amortized analysis paths the shortestpath tree theorem and general labeling method topological scanning, Dijkstra's algorithm, breadthfirst scanning Max Min flow problem cost flow problem matching general concepts and augmenting path method, including analysis negative cycles and cost reduction, leastcost augmenting path Bipartite Hashing finding matchings using flows, augmenting paths in bipartite graphs dleft vs. conventional hash table, how implementation affects cache performance, Bloom filters 2 1 Engineering Min spanning trees and heaps State the greedy method How is Prim's algorithm an implementation of the greedy method? Given a connected graph with 100 vertices and 300 edges. Suppose we apply the red rule 201 times, what can we say about the uncolored edges? Given a graph and partial mst computed by Prim's algorithm
what vertices are in the heap? for each vertex u in the heap, what is cheap(u), key(u)? In a 3heap with 100 nodes, how many are 3 levels down from the root? How many siblings does "last node" have? 3 Engineering Fibonacci Heaps Can a Fheap with 1000 nodes have a node with rank 10? Can it have a node with rank 16? Note: logn 1.44 lg n For a specific Fheap, how many credits are needed to maintain the credit invariant? For an Fheaps data structure with 1000 nodes how many credits are needed after initialization? after 500 melds? Give an upper bound on number of credits Consider a single Fheap on 1000 nodes Give a bound on the number of new trees created in the first step of a deletemin bound # of steps in the second part for which there is no collision bound # of credits needed to maintain invariant 4 2 Engineering Shortest paths State the shortest path tree theorem suppose dist(u)=100, dist(v)=110 and length(u,v)=20 and the SPT edge joining v to its parent is removed from the graph; can you bound length of new shortest path to v? Why is scanning & labeling a variant of labeling method? How do we know it never misses an edge? Let Nk(G,s) be the number of vertices in G that have a shortest path from s with k edges after 550 steps of the breadthfirst scanning algorithm, how many vertices have dist(u)=shortest path distance if n=100? Given a digraph with positive edge weights, describe an algorithm to detect if it has a cycle for which the product of the edge weights is <1 what is its running time?
5 Engineering Max flow problem What Given properties define a valid flow function? a flow, what is its residual graph? under what conditions is the sum of two flows a flow? identify a shortest augmenting path, how does the residual graph change when you add flow? In shortest augmenting path algorithm, if leveli(t) =levelj(t), how many times can edge (u,v) be included in an augmenting path in steps i+1 to j Describe an algorithm to determine the edge connectivity of a graph (# of edges that can be deleted without disconnecting graph) Flow graphs with lower bounds on flows finding a feasible flow, turning feasible flow into max flow
6 3 Engineering Mincost flows Given a graph and intermediate state for leastcost augmenting path algorithm show residual graph and leastcost spanning tree; find next path and update state; does new residual graph have negative cycles? In a graph with 200 vertices, 700 edges and a maxflow value of 137, a mincost, max flow with value 52 and a maxcost max flow with value 300 bound # of steps used by mincost augmenting path algorithm bound # of neg. cycles found by cycle reduction method 7 Engineering Matchings Given a matching, show how to find a larger matching; or a matching of larger weight Determine if a given matching is max weight. Give a general method for determining if a matching in a bipartite graph has max weight Give an algorithm to find a max wt matching in a bipartite graph using the cycle reduction algorithm for mincost flows Identify three augmenting paths with respect to a specific matching in a given graph For a given graph and matching, show a tree with root x that might be built by augmenting path algorithm; make tree as large as you can 8 4 Engineering Hashing Given a chart showing the distribution of expected # of buckets with specified length in conventional hash table what's the length of the longest bucket you're likely to see? what's the maximum # of cache misses you can expect? what's the chance that a random lookup hits a bucket with at least 5 items? In a 4left hash table using 4 arrays of buckets with 8 items per bucket and a separate (key,value) table assuming no part of the hash table is in cache initially, what is the minimum # of cache misses during a successful search? what is the maximum you would normally expect to see? what is the absolute maximum you could see? Give upper bound on false positive prob. for a 1 Kbyte Bloom filter with 5 hash functions, storing 1000 items 9 5 ...
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This note was uploaded on 03/08/2012 for the course CSE 542 taught by Professor Jonturner during the Spring '12 term at Washington State University .
 Spring '12
 JonTurner
 Computer Science

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