CE 221 Data Structures and Algorithms Chapter 9 : Graphs Part I - - PowerPoint PPT Presentation

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CE 221 Data Structures and Algorithms Chapter 9 : Graphs Part I - - PowerPoint PPT Presentation

Feb 25, 2023 130 likes •420 views

CE 221 Data Structures and Algorithms Chapter 9 : Graphs Part I (Topological Sort & Shortest Path Algorithms) Text: Read Weiss, 9.1 9.3 Izmir University of Economics 1 Definitions - I A graph G=(V, E) consists of a set of

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Chapter 9 : Graphs Part I (Topological Sort & Shortest Path Algorithms)

CE 221 Data Structures and Algorithms

Izmir University of Economics

Text: Read Weiss, §9.1 – 9.3

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Definitions - I

  • A graph G=(V, E) consists of a set of vertices,

V, and a set of edges, E.

  • Each edge is a pair (v, w), where v, w є V.
  • If the pair is ordered then G is directed

(digraph).

  • Vertex w is adjacent to v iff (v, w) є E.
  • In an undirected graph with edge (v, w), w is

adjacent to v and v is adjacent to w.

  • Sometimes and edge has a third component,

weight or cost.

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Definitions - II

  • A path in a graph is w1, w2,...,wN such that

(wi, wi+1) є E for 1≤i<N. The length of such

a path is the number of edges on the path. If a path from a vertex to itself contains no edges, then the path length is zero. If G contains an edge (v, v), then the path v, v is called a loop.

  • A simple path is a path such that all

vertices are distinct, except that the first and the last could be the same.

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Definitions - III

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  • A cycle in a directed graph is a path of length at

least 1 such that w1=wN. This cycle is simple if the path is simple. For undirected graphs, the edges are required to be distinct (Why?).

  • A directed graph is acyclic if it has no cycles (DAG).
  • An undirected graph is connected if there is a path

from every vertex to every other vertex. A directed graph with this property is called strongly

  • connected. If directed graph is not, but underlying

undirected graph is, it is weakly connected. A complete graph is a graph in which there is an edge between every pair of vertices.

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Representation of Graphs - I

  • One simple way is to use a two-

dimensional array (adjacency matrix representation). If vertices are numbered starting at 1, A[u][v]=true if (u, v) є E. Space requirement is Θ(|V|2).

  • If the graph is not dense (sparse),

adjacency lists may be used. The space requirement is O(|E|+|V|).

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Representation of Graphs - II

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Topological Sort - I

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  • A topological sort is an ordering of vertices

in a DAG, such that if there is path from vi to vj, then vj appears after vi in the ordering.

  • A simple algorithm to find a topological
  • rdering is first to find any vertex with no

incoming edges. We can then print this vertex, and remove it, along with its edges. Then apply the same strategy to the rest of the graph. To formalize this, define the indegree of a vertex v as the number of edges (u, v).

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Topological Sort – Initial Attempt

  • running time of

the algorithm is O(|V|2).

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  • We can remove the inefficiency by

keeping all the unassigned vertices

  • f indegree 0 in a special data

structure (queue or stack). When a new vertex with degree zero is needed, it is returned by removing

  • ne from the queue, and when the

indegrees of adjacent vertices are decremented, they are inserted into the queue if the indegree falls to

  • zero. The running time is O(|E|+|V|)

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Topological Sort – A Better Algorithm

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Homework Assignments

  • 9.1, 9.2, 9.3, 9.4, 9.38
  • You are requested to study and solve the
  • exercises. Note that these are for you to

practice only. You are not to deliver the results to me.

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Shortest-Path Algorithms

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  • The input is a weighted graph: associated

with each edge (vi, vj) is a cost ci,j. The cost of a path v1v2...vN is ∑ci,i+1 for i in [1..N-1]. This is weighted path length, the unweighted path length on the other hand is merely the number of edges on the path, namely, N-1.

  • Single-source Shortest-Path Problem:

Given as input a weighted graph G=(V, E), and a distinguished vertex, s, find the shortest weighted path from s to every

  • ther vertex in G.
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Negative Cost Cycles

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  • In the graph to the left, the shortest path from v1 to v6 has a

cost of 6 and the path itself is v1v4v7v6. The shortest unweighted path has 2 edges.

  • In the graph to the right, we have a negative cost. The path

from v5 to v4 has cost 1, but a shorter path exists by following the loop v5v4v2v5v4 which has cost -5. This path is still not the shortest, because we could stay in the loop arbitrarily long.

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Shortest Path Length: Problems

We will examine 4 algorithms to solve four versions of the problem 1.Unweighted shortest path  O(|E|+|V|) 2.Weighted shortest path without negative edges  O(|E|log|V|) using queues 3.Weighted shortest path with negative edges  O(|E| . |V|) 4.Weighted shortest path of acyclic graphs  linear time

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Unweighted Shortest Paths

  • Using some vertex, s, which is an input

parameter, find the shortest path from s to all

  • ther vertices in an unweighted graph. Assume

s=v3.

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Unweighted Shortest Paths

  • Algorithm: find vertices that are at

distance 1, 2, ... N-1 by processing vertices in layers (breadth-first search)

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Unweighted Shortest Paths

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Unweighted Shortest Paths

  • Complexity O(|V|2)

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Unweighted Shortest Paths - Improvement

  • At any point in time there

are only two types of unknown vertices that have dv≠∞. Some have dv = currDist and the rest have dv = currDist +1.

  • We can make use of a

queue data structure.

  • O(|E|+|V|)

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Weighted Shortest Path Dijkstra’s Algorithm

  • With weighted shortest path,distance dv is
  • tentative. It turns out to be the shortest path

length from s to v using only known vertices as intermediates.

  • Greedy algorithm: proceeds in stages doing the

best at each stage. Dijkstra’s algorithm selects a vertex v with smallest dv among all unknown vertices and declares it known. Remainder of the stage consists of updating the values dw for all edges (v, w).

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Dijkstra’s Algorithm - Example

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► ► ►

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Dijkstra’s Algorithm - Example

  • A proof by contradiction will show that this

algorithm always works as long as no edge has a negative cost.

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► ► ► ►

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Dijkstra’s Algorithm - Pseudocode

  • If the vertices are

sequentially scanned to find minimum dv, each phase will take O(|V|) to find the minimum, thus O(|V|2) over the course

  • f the algorithm.
  • The time for updates is

constant and at most

  • ne update per edge for

a total of O(|E|).

  • Therefore the total time

spent is O(|V|2+|E|).

  • If the graph is dense,

OPTIMAL.

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Dijkstra’s Algorithm-What if the graph is sparse?

  • If the graph is sparse |E|=θ(|V|), algorithm is too
  • slow. The distances of vertices need to be kept

in a priority queue.

  • Selection of vertex with minimum distance via

deleteMin, and updates via decreaseKey

  • peration. Hence; O(|E|log|V|+|V|log|V|)
  • find operations are not supported, so you need

to be able to maintain locations of di in the heap and update them as they change.

  • Alternative: insert w and dw with every update.

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Graphs with negative edge costs

  • Dijkstra’s algorithm does not work with

negative edge costs. Once a vertex u is known, it is possible that from some other unknown vertex v, there is a path back to u that is very negative.

  • Algorithm: A combination of weighted and

unweighted algorithms. Forget about the concept of known vertices.

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Graphs with negative edge costs - I

  • O(|E|*|V|). Each vertex

can dequeue at most O(|V|) times. (Why? Algorithm computes shortest paths with at most 0, 1, ..., |V|-1 edges in this order). Hence, the result!

  • If negative cost cycles,

then each vertex should be checked to have been dequeued at most |V| times.

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Acyclic Graphs

  • If the graph is known to be acyclic, the
  • rder in which vertices are declared

known, can be set to be the topological

  • rder.
  • Running time = O(|V|+|E|)
  • This selection rule works because when a

vertex is selected, its distance can no longer be lowered, since by topological

  • rdering rule it has no incoming edges

emanating from unknown nodes.

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Homework Assignments

  • 9.5, 9.7, 9.10, 9.40, 9.42, 9.44, 9.46, 9.47,

9.52

  • You are requested to study and solve the
  • exercises. Note that these are for you to

practice only. You are not to deliver the results to me.

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