graph theory Chapter 9 Graph Algorithms - William & …tadavis/cs303/ch09sm.pdf · Chapter 9 Graph Algorithms Introduction 2 graph theory ... 9 in a directed graph ... A. Church and

Chapter 9Graph Algorithms

Introduction

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graph theoryuseful in practicerepresent many real-life problemscan be if not careful with data structures

Definitions

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an undirected graph is a finite set of____________ together with a set of ___________an edge is a pair , where and are vertices

this definition allows____________, or edges that connect vertices tothemselvesparallel edges, or multiple edges that connect the samepair of vertices

a graph without self-loops is a simple grapha graph with parallel edges is sometimes called a multigraph

Definitions

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two vertices are adjacent if there is an between themthe edge is said to be incident to the two vertices

if there are no parallel edges, the degree of a vertex is thenumber of edges to it

self-loops add only to the degree

a subgraph of a graph is a subset of 's edges togetherwith the incident vertices

Definitions

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a path in a graph is a sequence of vertices connected by____________

a simple path is a path with no ______________ vertices,except possibly the first and last

a cycle is a path of at least one edge whose first and last___________ are the same

a simple cycle is a cycle with no repeated edges ofvertices other than the first and last

the length of a path is the number of in the path

Definitions

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a graph is connected if every vertex is connected to everyother vertex by a through the graph

a connected component of a graph is a maximalconnected subgraph of : if is a subset of and is aconnected subgraph of , then

a graph that is not connected consists of a set of connected_______________

a graph without cycles is called acyclic

Definitions

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a tree is a connected, acyclic grapha forest is a set of treesa spanning tree of a connected graph is a subgraph that is atree and also contains all of the graph's _____________a spanning forest of a graph is the union of spanning treesof its connected components

Definitions

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if is the number of vertices and is the number ofedges, then, in a graph without self-loops and paralleledges, there are possible ___________a graph is complete if there is an edge between every________ of verticesthe density of a graph refers to the proportion of possiblepairs of vertices that are connecteda sparse graph is one for whicha dense graph is a graph that is not ____________a bipartite graph is one whose vertices can be divided intotwo sets so that every vertex in one set is connected to atleast one vertex in the other set

Definitions

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in a directed graph or digraph the pairs indicatingedges are : the edge goes from(the tail) to (the head)since edges have a direction, we use the notation todenote an edge from toedges in digraphs are frequently called arcsthe of a vertex is the number of arcs

(i.e., the number of arcs coming into ), while theoutdegree of is the number of arcs (i.e., thenumber of arcs exiting )we will call a source if its indegree is ____an aborescence is a directed graph with a distinguishedvertex (the root) such that for every other vertex there isa unique directed path from to

Definitions

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in a directed graph, two vertices and are ____________connected if there is a directed path from to and adirected path from toa digraph is strongly connected if all its vertices are stronglyconnectedif a digraph is not strongly connected but the underlyingundirected graph is connected, then the digraph is calledweakly connecteda weighted graph has weights or costs associated with each___________

weighted graphs can be directed or undirecteda road map with mileage is the prototypical example

Example

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airport connections

http://allthingsgraphed.com/2014/08/09/us-airlines-graph/

Graph Representation

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two concerns: and ___________

similarthe following graph has 7 vertices and 12 edges


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adjacency matrix2D matrix where an element is 1 if and 0otherwise


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adjacency matrixalternatively, could use costs or for _____________not efficient if the graph is (number of edges small)

matrixe.g., street map with 3,000 streets results in intersectionmatrix with 9,000,000 elements

adjacency liststandard way to represent graphsundirected graph edges appear in listmore efficient if the graph is sparse (number of edgessmall)

matrix


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adjacency listfor each vertex, keep a list of vertices


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adjacency list alternativefor each vertex, keep a of adjacent vertices

Topological Sort

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a directed acyclic graph (DAG) is a digraph with no directedcycles

a DAG always has at least one __________

topological sortan ordering of the vertices in a directed graph such that ifthere is a path from v to w, then v appears w inthe orderingnot possible if graph has a ___________

Topological Sort

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example directed acyclic graph

Topological Sort

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topological sortdetermine the indegree for every

place all source vertices in a queuewhile there remains a

find a vertexappend the source vertex to the topological sortdelete the source and its adjacent fromupdate the of the remaining vertices inplace any new source vertices in the queue

when no vertices remain, we have our ordering, or, if we aremissing vertices from the output list, the graph has notopological sort

Topological Sort

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since finding vertex with 0 indegree must look at _______vertices, and this is performed times,

Topological Sort

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instead, we can keepall the vertices withindegree 0 in a ______and choose from there

Topological Sort

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adjacency list alternativefor each , keep a vector of adjacent vertices

Shortest-Path Algorithms

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shortest-path problemsinput is a weighted graph with a on each edgeweighted path length:

single-source shortest-path problemgiven as input a weighted graph, and a_______________ vertex , find the shortest weightedpath from to every other vertex in


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exampleshortest weighted path from to has cost of _____no path from to


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edges can cause problemspath from to has cost of 1, but a shorter path exists byfollowing the negative loop, which has cost -5shortest paths thus _______________


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many examples where we want to find shortest pathsif vertices represent computers and edges connections, thecost represents costs, delay costs, orcombination of costsif vertices represent airports and edges costs to travelbetween them, shortest path is route

we find paths from one vertex to others since noalgorithm exists that finds shortest path from one vertex toone other faster


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four problemsunweighted shortest-pathweighted shortest-path with no negative edgesweighted shortest-path with negative edgesweighted shortest-path in acyclic graphs

Unweighted Shortest Paths

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unweighted shortest-pathfind shortest paths from to all other verticesonly concerned with number of in path

we will not actually record the path ________________


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examplestart with


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examplemark 0 length to


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examplemark 1 length for and


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examplemark 2 length for and


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examplefinal path assignments


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searching an unweighted shortest-path uses a_________________ search

processes vertices in layers, according to _____________begins with initializing path lengths

a vertex will be marked when the shortest pathto it is found


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with this algorithmpath can be printedrunning time:bad case

can reduce time by keeping vertices that are unknown__________________ from those knownnew running time:


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Algorithm

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weighted shortest-path algorithmmore difficult, but ideas from algorithmcan be usedkeep information as before for each vertex

knownset if using onlyknown vertices

the last vertex to cause a change toalgorithm

does what appears to be best thing at each stagee.g., counting money: count quarters first, then dimes,nickels, pennies

gives change with least number of coins

Algorithm

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pseudocodeassumption: no weightsorigin is given

Algorithm

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pseudocode (cont.)

Algorithm

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example: start at

Algorithm

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example

Algorithm

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example

,

Algorithm

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example stages shown on the graph

Algorithm

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example stages shown on the graph (cont.)

Algorithm: Correctness

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correctness

Algorithm

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complexitysequentially scanning vertices to find minimum takes

, which results in overallat most one update per edge, for a total of

if graph is dense, with , algorithm is close to________________if graph is sparse, with , algorithm is too__________

distances could be kept in a queue thatreduces running time to

Algorithm

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implementationinformation for each vertex

Algorithm

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implementation (cont.)path can be printed recursively

Algorithm

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implementation (cont.)

Graphs with Negative Edges

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try to apply algorithm to graph with edges

Graphs with Negative Edges

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possible solution: a delta value to all weights suchthat none are negative

calculate path on new graphapply path to original graphdoes not work: longer paths become weightier

combination of algorithms for weighted graphs andunweighted graphs can work

drastic in running time:

All-Pairs Shortest Paths

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given a weighted digraph, find the shortest paths between_____ vertices in the graphone approach: apply Dijkstra's algorithm ________________

results inanother approach: apply Floyd-Warshall algorithm

uses programmingalso results in

Minimum Spanning Tree

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assumptionsgraph is ______________edge weights are not necessarily Euclidean distancesedge weights need not be all the sameedge weights may be zero or ______________

minimum spanning tree (MST)also called minimum-weight spanning tree of a weightedgraphspanning tree whose weight (the sum of the weights of theedges in the tree) is the among all spanningtrees


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example


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two algorithms to find the minimum spanning tree

R. C. Prim, Shortest Connection Networks and SomeGeneralizations, Bell System Technical Journal (1957)

AlgorithmJ. B. Kruskal, Jr., On the Shortest Spanning Subtree of aGraph and the Traveling Salesman Problem, Proc. of theAmerican Mathematical Society (1956)


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grows tree in successive stages


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, , & , , &


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runs on graphsrunning time: without heaps, which is optimal for___________ graphsrunning time: using binary heaps, which is goodfor graphs


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algorithmcontinually select the edges in order of ______________weightaccept the edge if it does not cause a withalready accepted edges


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algorithm: example


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algorithm: example 2


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algorithm: example 2 (cont.)


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time complexity: with proper choice and use ofdata structuresin the worst case, , so the worst-case timecomplexity is

NP-Complete Problems

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What problems can we solve algorithmically? which problemsare easy? which problems are hard?

Eulerian circuit: Given a vertex s, start at s and find a cyclethat visits every exactly once

easy: solvable in using depth-first search

Hamiltonian circuit: Given a vertex s, start at s and find acycle that visits each remaining exactly once

really, really hard!


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halting problemin 1936, A. Church and A. Turing independently proved thenon-solvability of the halting problem:

is there an algorithm terminates(p,x) that takes anarbitrary program p and input x and returns True if pterminates when given input x and False otherwise?

difficult: try to run it on _____________


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halting problemsuppose we had such an algorithm terminates(p,x)create a new program:

program evil (z) {1: if terminates(z,z) goto 1

}

program evil() terminates if and only if the program zdoes not terminate when given its own code as input

no such algorithm can exist


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decision problemhas a yes or no answerundecidable if it is to construct a singlealgorithm that will solve all instances of the problemthe halting problem is _______________


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the class Pset of problems for which there exists a _______________time algorithm for their solutionthe runtime is bounded by a polynomial function of the_________ of the problem

the class NPset of decision problems for which the certification of acandidate ______________ as being correct can beperformed in polynomial timenon-deterministic polynomial time


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the class NPfor problems in NP, certifying a solution may not be difficult,but a solution may be very difficultexample: Hamiltonian circuit

given a graph G, is there a simple cycle in G thatincludes every ?given a candidate solution, we can check whether it is asimple cycle in time , simply by thepathhowever, finding a Hamiltonian circuit is hard!


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reductionsproblem A to problem B if the solvabilityof B implies the solvability of A

if A is reducible to B, then B is at least as hard to solveas A

in the context of algorithms, reducibility means analgorithm that solves B can be into analgorithm to solve A

example: if we can sort a set of numbers, we can findthe median, so finding the median reduces to sorting


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reductionsproblem A can be reduced to B if we cansolve problem A using an algorithm for problem B suchthat the cost of solving A is

cost of solving B + a polynomial function of the problem size

example: once we have sorted an array ofnumbers, we can find the median in timeby computing and accessing


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reductionsdecision version of traveling salesperson problem (TSP):

given a complete weighted graph and an integer ,does there exist a simple cycle that allvertices (tour) with total weight ?

clearly, this is in NPHamiltonian circuit: given a graph , find asimple cycle that visits all the vertices

construct a new graph with the same vertices asbut which is ; if an edge in is in ,give it weight 1; otherwise, give it weight 2

construction requires workapply to see if there exists a tour with totalweight


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reductions


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NP-completea problem A is NP-complete if it is in NP and all otherproblems in NP can be to A in polynomialtime

Boolean satisfiablity (SAT): given a set of N booleanvariables and M logical statements built from thevariables using and and not, can you choose values forthe variables so that all the statements are ?

SAT is NP-complete


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NP-completeif we restrict attention to sets of boolean statementsinvolving variables, the problem is known as 3-SAT

3-SAT is NP-completeso, if you can solve 3-SAT in polynomial time, you cansolve problems in NP in polynomial timemeanwhile, 2-SAT is solvable in time!


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NP-complete problemstraveling salespersonbin packingknapsackgraph coloringlongest-path


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NP-hard problemsa problem A is NP-hard if there exists a polynomial-timereduction from an NP-complete problem to Aan NP-hard problem is at as hard as an NP-complete problem

versions of NP-complete problems aretypically NP-hard

optimization version of TSP: given a weighted graph,find a cost Hamiltonian circuitif we can solve TSP, we can solve Hamiltonian circuit

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: 0.8, 0.2: 0.7, 0.3: 0.5, 0.4, 0.1

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graph theory Chapter 9 Graph Algorithms - William & …tadavis/cs303/ch09sm.pdf · Chapter 9 Graph Algorithms Introduction 2 graph theory ... 9 in a directed graph ... A. Church and