There are two common ways to traverse a graph, BFS and DFS. Considering a Tree (or Graph) of huge height and width, both BFS and DFS are not very efficient due to following reasons.
- DFS first traverses nodes going through one adjacent of root, then next adjacent. The problem with this approach is, if there is a node close to root, but not in first few subtrees explored by DFS, then DFS reaches that node very late. Also, DFS may not find shortest path to a node (in terms of number of edges).
- BFS goes level by level, but requires more space. The space required by DFS is O(d) where d is depth of tree, but space required by BFS is O(n) where n is number of nodes in tree (Why? Note that the last level of tree can have around n/2 nodes and second last level n/4 nodes and in BFS we need to have every level one by one in queue).
IDDFS combines depth-first search’s space-efficiency and breadth-first search’s fast search (for nodes closer to root).
How does IDDFS work?
IDDFS calls DFS for different depths starting from an initial value. In every call, DFS is restricted from going beyond given depth. So basically we do DFS in a BFS fashion.
Algorithm:
// Returns true if target is reachable from
// src within max_depth
bool IDDFS(src, target, max_depth)
for limit from 0 to max_depth
if DLS(src, target, limit) == true
return true
return false
bool DLS(src, target, limit)
if (src == target)
return true;
// If reached the maximum depth,
// stop recursing.
if (limit <= 0)
return false;
foreach adjacent i of src
if DLS(i, target, limit?1)
return true
return false
An important thing to note is, we visit top level nodes multiple times. The last (or max depth) level is visited once, second last level is visited twice, and so on. It may seem expensive, but it turns out to be not so costly, since in a tree most of the nodes are in the bottom level. So it does not matter much if the upper levels are visited multiple times. Below is implementation of above algorithm
C++
// C++ program to search if a target node is reachable from // a source with given max depth. #include<bits/stdc++.h> using namespace std; // Graph class represents a directed graph using adjacency // list representation. class Graph { int V; // No. of vertices // Pointer to an array containing // adjacency lists list<int> *adj; // A function used by IDDFS bool DLS(int v, int target, int limit); public: Graph(int V); // Constructor void addEdge(int v, int w); // IDDFS traversal of the vertices reachable from v bool IDDFS(int v, int target, int max_depth); }; Graph::Graph(int V) { this->V = V; adj = new list<int>[V]; } void Graph::addEdge(int v, int w) { adj[v].push_back(w); // Add w to v’s list. } // A function to perform a Depth-Limited search // from given source 'src' bool Graph::DLS(int src, int target, int limit) { if (src == target) return true; // If reached the maximum depth, stop recursing. if (limit <= 0) return false; // Recur for all the vertices adjacent to source vertex for (auto i = adj[src].begin(); i != adj[src].end(); ++i) if (DLS(*i, target, limit-1) == true) return true; return false; } // IDDFS to search if target is reachable from v. // It uses recursive DFSUtil(). bool Graph::IDDFS(int src, int target, int max_depth) { // Repeatedly depth-limit search till the // maximum depth. for (int i = 0; i <= max_depth; i++) if (DLS(src, target, i) == true) return true; return false; } // Driver code int main() { // Let us create a Directed graph with 7 nodes Graph g(7); g.addEdge(0, 1); g.addEdge(0, 2); g.addEdge(1, 3); g.addEdge(1, 4); g.addEdge(2, 5); g.addEdge(2, 6); int target = 6, maxDepth = 3, src = 0; if (g.IDDFS(src, target, maxDepth) == true) cout << "Target is reachable from source " "within max depth"; else cout << "Target is NOT reachable from source " "within max depth"; return 0; } |
Java
// Java program to search if a target node is reachable from// a source with given max depth.import java.util.*;// Graph class represents a directed graph using adjacency// list representation.class Graph { int V; // No. of vertices // Pointer to an array containing // adjacency lists LinkedList<Integer> adj[]; // A function used by IDDFS boolean DLS(int v, int target, int limit) { if (v == target) return true; // If reached the maximum depth, stop recursing. if (limit <= 0) return false; // Recur for all the vertices adjacent to source // vertex for (int i : adj[v]) if (DLS(i, target, limit - 1)) return true; return false; } public Graph(int v) { V = v; adj = new LinkedList[v]; for (int i = 0; i < v; ++i) adj[i] = new LinkedList(); } void addEdge(int v, int w) { adj[v].add(w); // Add w to v's list. } // IDDFS to search if target is reachable from v. // It uses recursive DFSUtil(). boolean IDDFS(int src, int target, int max_depth) { // Repeatedly depth-limit search till the // maximum depth. for (int i = 0; i <= max_depth; i++) if (DLS(src, target, i)) return true; return false; }}// Driver codeclass Main { public static void main(String[] args) { // Let us create a Directed graph with 7 nodes Graph g = new Graph(7); g.addEdge(0, 1); g.addEdge(0, 2); g.addEdge(1, 3); g.addEdge(1, 4); g.addEdge(2, 5); g.addEdge(2, 6); int target = 6, maxDepth = 3, src = 0; if (g.IDDFS(src, target, maxDepth)) System.out.println( "Target is reachable from source " + "within max depth"); else System.out.println( "Target is NOT reachable from source " + "within max depth"); }} |
Python
# Python program to print DFS traversal from a given# given graphfrom collections import defaultdict# This class represents a directed graph using adjacency# list representationclass Graph: def __init__(self,vertices): # No. of vertices self.V = vertices # default dictionary to store graph self.graph = defaultdict(list) # function to add an edge to graph def addEdge(self,u,v): self.graph[u].append(v) # A function to perform a Depth-Limited search # from given source 'src' def DLS(self,src,target,maxDepth): if src == target : return True # If reached the maximum depth, stop recursing. if maxDepth <= 0 : return False # Recur for all the vertices adjacent to this vertex for i in self.graph[src]: if(self.DLS(i,target,maxDepth-1)): return True return False # IDDFS to search if target is reachable from v. # It uses recursive DLS() def IDDFS(self,src, target, maxDepth): # Repeatedly depth-limit search till the # maximum depth for i in range(maxDepth): if (self.DLS(src, target, i)): return True return False# Create a graph given in the above diagramg = Graph (7);g.addEdge(0, 1)g.addEdge(0, 2)g.addEdge(1, 3)g.addEdge(1, 4)g.addEdge(2, 5)g.addEdge(2, 6)target = 6; maxDepth = 3; src = 0if g.IDDFS(src, target, maxDepth) == True: print ("Target is reachable from source " + "within max depth")else : print ("Target is NOT reachable from source " + "within max depth")# This code is contributed by Neelam Pandey |
C#
using System;using System.Collections.Generic;// Graph class represents a directed graph using adjacency// list representation.class Graph { private int V; // No. of vertices // Pointer to an array containing adjacency lists private List<int>[] adj; // Constructor public Graph(int V) { this.V = V; adj = new List<int>[ V ]; for (int i = 0; i < V; ++i) adj[i] = new List<int>(); } // Add w to v's list public void addEdge(int v, int w) { adj[v].Add(w); } // A function used by IDDFS private bool DLS(int v, int target, int limit) { if (v == target) return true; // If reached the maximum depth, stop recursing. if (limit <= 0) return false; // Recur for all the vertices adjacent to source // vertex foreach( int i in adj[v]) if (DLS(i, target, limit - 1)) return true; return false; } // IDDFS to search if target is reachable from v. // It uses recursive DLS(). public bool IDDFS(int src, int target, int max_depth) { // Repeatedly depth-limit search till the maximum // depth. for (int i = 0; i <= max_depth; i++) if (DLS(src, target, i)) return true; return false; }}// Driver codeclass Program { static void Main(string[] args) { // Let us create a Directed graph with 7 nodes Graph g = new Graph(7); g.addEdge(0, 1); g.addEdge(0, 2); g.addEdge(1, 3); g.addEdge(1, 4); g.addEdge(2, 5); g.addEdge(2, 6); int target = 6, maxDepth = 3, src = 0; if (g.IDDFS(src, target, maxDepth)) Console.WriteLine( "Target is reachable from source within max depth"); else Console.WriteLine( "Target is NOT reachable from source within max depth"); }}// This code is contributed by lokeshpotta20. |
Javascript
<script> /* Javascript program to search if a target node is reachable from a source with given max depth.*/ // Graph class represents a directed graph using adjacency // list representation. class Graph { constructor(V) { this.V = V; this.adj = new Array(V); for(let i = 0; i < V; i++) this.adj[i] = []; } addEdge(v, w) { this.adj[v].push(w); // Add w to v’s list. } // A function to perform a Depth-Limited search // from given source 'src' DLS(src, target, limit) { if (src == target) return true; // If reached the maximum depth, stop recursing. if (limit <= 0) return false; // Recur for all the vertices adjacent to source vertex for (let i of this.adj[src].values()) { if (this.DLS(i, target, limit-1) == true) return true; } return false; } // IDDFS to search if target is reachable from v. // It uses recursive DFSUtil(). IDDFS(src, target, max_depth) { // Repeatedly depth-limit search till the // maximum depth. for (let i = 0; i <= max_depth; i++) { if (this.DLS(src, target, i) == true) return true; } return false; } } // Driver code // Let us create a Directed graph with 7 nodes g = new Graph(7); g.addEdge(0, 1); g.addEdge(0, 2); g.addEdge(1, 3); g.addEdge(1, 4); g.addEdge(2, 5); g.addEdge(2, 6); let target = 6, maxDepth = 3, src = 0; if (g.IDDFS(src, target, maxDepth) == true) document.write("Target is reachable from source within max depth"); else document.write("Target is NOT reachable from source within max depth"); // This code is contributed by cavi4762.</script> |
Output
Target is reachable from source within max depth
Illustration: There can be two cases:
- When the graph has no cycle: This case is simple. We can DFS multiple times with different height limits.
- When the graph has cycles. This is interesting as there is no visited flag in IDDFS.
Time Complexity: Suppose we have a tree having branching factor ‘b’ (number of children of each node), and its depth ‘d’, i.e., there are bd nodes. In an iterative deepening search, the nodes on the bottom level are expanded once, those on the next to bottom level are expanded twice, and so on, up to the root of the search tree, which is expanded d+1 times. So the total number of expansions in an iterative deepening search is-
(d)b + (d-1)b2 + .... + 3bd-2 + 2bd-1 + bd That is, Summation[(d + 1 - i) bi], from i = 0 to i = d Which is same as O(bd)
After evaluating the above expression, we find that asymptotically IDDFS takes the same time as that of DFS and BFS, but it is indeed slower than both of them as it has a higher constant factor in its time complexity expression. IDDFS is best suited for a complete infinite tree
Complexity:
Time: O(b^d)
Space: O(d)
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