As per my understanding, I have calculated time complexity of Dijkstra Algorithm as big-O notation using adjacency list given below. It didn't come out as it was supposed to and that led me to understand it step by step.
Each vertex can be connected to (V-1) vertices, hence the number of adjacent edges to each vertex is V - 1. Let us say E represents V-1 edges connected to each vertex.
Finding & Updating each adjacent vertex's weight in min heap is O(log(V)) + O(1) or O(log(V)).
Hence from step1 and step2 above, the time complexity for updating all adjacent vertices of a vertex is E*(logV). or E*logV.
Hence time complexity for all V vertices is V * (E*logV) i.e O(VElogV).
But the time complexity for Dijkstra Algorithm is O(ElogV). Why?
Dijkstra's shortest path algorithm is O(ElogV) where:
V is the number of vertices
E is the total number of edges
Your analysis is correct, but your symbols have different meanings! You say the algorithm is O(VElogV) where:
V is the number of vertices
E is the maximum number of edges attached to a single node.
Let's rename your E to N. So one analysis says O(ElogV) and another says O(VNlogV). Both are correct and in fact E = O(VN). The difference is that ElogV is a tighter estimation.
Adding a more detailed explanation as I understood it just in case:
O(for each vertex using min heap: for each edge linearly: push vertices to min heap that edge points to)
V = number of vertices
O(V * (pop vertex from min heap + find unvisited vertices in edges * push them to min heap))
E = number of edges on each vertex
O(V * (pop vertex from min heap + E * push unvisited vertices to min heap)). Note, that we can push the same node multiple times here before we get to "visit" it.
O(V * (log(heap size) + E * log(heap size)))
O(V * ((E + 1) * log(heap size)))
O(V * (E * log(heap size)))
E = V because each vertex can reference all other vertices
O(V * (V * log(heap size)))
O(V^2 * log(heap size))
heap size is V^2 because we push to it every time we want to update a distance and can have up to V comparisons for each vertex. E.g. for the last vertex, 1st vertex has distance 10, 2nd has 9, 3rd has 8, etc, so, we push each time to update
O(V^2 * log(V^2))
O(V^2 * 2 * log(V))
O(V^2 * log(V))
V^2 is also a total number of edges, so if we let E = V^2 (as in the official naming), we will get the O(E * log(V))
let n be the number of vertices and m be the number of edges.
Since with Dijkstra's algorithm you have O(n) delete-mins and O(m) decrease_keys, each costing O(logn), the total run time using binary heaps will be O(log(n)(m + n)). It is totally possible to amortize the cost of decrease_key down to O(1) using Fibonacci heaps resulting in a total run time of O(nlogn+m) but in practice this is often not done since the constant factor penalties of FHs are pretty big and on random graphs the amount of decrease_keys is way lower than its respective upper bound (more in the range of O(n*log(m/n), which is way better on sparse graphs where m = O(n)). So always be aware of the fact that the total run time is both dependent on your data structures and the input class.
In dense(or complete) graph, E logV > V^2
Using linked data & binary heap is not always best.
That case, I prefer to use just matrix data and save minimum length by row.
Just V^2 time needed.
In case, E < V / logV.
Or, max edges per vertex is less than some constant K.
Then use binary heap.
I find it easier to think at this complexity in the following way:
The nodes are first inserted in a priority queue and the extracted one by one leading to O(V log V).
Once a node is extracted, we iterate through its edges and update the priority queue accordingly. Note that every edge is explored only once, moreover, updating the priority queue is O(log V), leading to an overall O(E log V).
TLDR. You have V extractions from the priority queue and E updates to the priority queue, leading to an overall O((V + E) log V).
Let's try to analyze the algorithm as given in CLRS book.
for each loop in line 7: for any vertex say 'u' the number of times the loop runs is equal to the number of adjacent vertices of 'u'.
The number of adjacent vertices for a node is always less than or equal to the total number of edges in the graph.
If we take V (because of while loop in line 4) and E (because of for each in line 7) and compute the complexity as VElog(V) it would be equivalent to assuming each vertex has E edges incident on it, but in actual there will be atmost or less than E edges incident on a single vertex. (the atmost E adjacent vertices for a single vertex case happens in case of a star graph for the internal vertex)
V:Number of Vertices,
E:Number of total_edges
Suppose the Graph is dense
The complexity would be O(V*logV) + O( (1+2+...+V)*logV)
1+2+....+(V-1) = (v)*(v+1)/2 ~ V^2 ~ E because the graph is dense
So the complexity would be O(ElogV).
The sum 1+2+...+ V refers to: For each vertex v in G.adj[u] but not in S
If you think about Q before a vertex is extracted has V vertices then it has V-1 then V-2
... then 1.
E is edges and V is vertices. Number of edges
(V *(V-1)) / 2
approximately
V ^ 2
So we can add maximum V^2 edges to the min heap. So sorting the elements in min heap will take
O(Log(V ^ 2))
Every time we insert a new element into min heap, we are going to sort. We will have E edges so we will be sorting E times. so total time complexity
O(E * Log(V ^ 2)= O( 2 * E * Log(V))
Omitting the constant 2:
O( E * Log(V))
Related
I would like to know, what would be the most efficient way (w.r.t., Space and Time) to solve the following problem:
Given an undirected Graph G = (V, E), a positive number N and a vertex S in V. Assume that every vertex in V has a cost value. Find the N highest cost vertices that is connected to S.
For example:
G = (V, E)
V = {v1, v2, v3, v4},
E = {(v1, v2),
(v1, v3),
(v2, v4),
(v3, v4)}
v1 cost = 1
v2 cost = 2
v3 cost = 3
v4 cost = 4
N = 2, S = v1
result: {v3, v4}
This problem can be solved easily by the graph traversal algorithm (e.g., BFS or DFS). To find the vertices connected to S, we can run either BFS or DFS starting from S. As the space and time complexity of BFS and DFS is same (i.e., time complexity: O(V+E), space complexity: O(E)), here I am going to show the pseudocode using DFS:
Parameter Definition:
* G -> Graph
* S -> Starting node
* N -> Number of connected (highest cost) vertices to find
* Cost -> Array of size V, contains the vertex cost value
procedure DFS-traversal(G,S,N,Cost):
let St be a stack
let Q be a min-priority-queue contains <cost, vertex-id>
let discovered is an array (of size V) to mark already visited vertices
St.push(S)
// Comment: if you do not want to consider the case "S is connected to S"
// then, you can consider commenting the following line
Q.push(make-pair(S, Cost[S]))
label S as discovered
while St is not empty
v = St.pop()
for all edges from v to w in G.adjacentEdges(v) do
if w is not labeled as discovered:
label w as discovered
St.push(w)
Q.push(make-pair(w, Cost[w]))
if Q.size() == N + 1:
Q.pop()
let ret is a N sized array
while Q is not empty:
ret.append(Q.top().second)
Q.pop()
Let's first describe the process first. Here, I run the iterative version of DFS to traverse the graph starting from S. During the traversal, I use a priority-queue to keep the N highest cost vertices that is reachable from S. Instead of the priority-queue, we can use a simple array (or even we can reuse the discovered array) to keep the record of the reachable vertices with cost.
Analysis of space-complexity:
To store the graph: O(E)
Priority-queue: O(N)
Stack: O(V)
For labeling discovered: O(V)
So, as O(E) is the dominating term here, we can consider O(E) as the overall space complexity.
Analysis of time-complexity:
DFS-traversal: O(V+E)
To track N highest cost vertices:
By maintaining priority-queue: O(V*logN)
Or alternatively using array: O(V*logV)
The overall time-complexity would be: O(V*logN + E) or O(V*logV + E)
A well-known 2-approximation for a Minimum Weighted Vertex Cover Problem is the one proposed by Clarkson:
Clarkson, Kenneth L. "A modification of the greedy algorithm for vertex cover." Information Processing Letters 16.1 (1983): 23-25.
Easy-to-read pseudo code of the algorithm can be found here see section 32.1.2.
The algorithm, according to the paper, has a runtime complexity of O(|E|*log|V|) where E is the set of edges and V the set of vertices. I'm not entirely sure how they get this result.
Let d(v) be the degree of vertex v in a graph, and w(v) be some weight function.
Excluding some of the technicalities from the algorithm, the algorithm looks like this:
while( |E| != 0){ //While there are still edges in the graph
Pick a vertex v \in V for which w(v)/d(v) is minimized;
for( u : (u,v) \in E){
update w(u);
...
}
delete v and all edges incident to it from the graph.
}
The outer loop produces the term |E| in the runtime complexity. That means that picking a vertex out of a list of vertices which minimizes some ratio can be done in log n time. As far as I can tell, finding a minimum value out of a list of values takes n-1 comparisons, not log n. Finally, the inner for loop runs for every neighbor of v, so yields a complexity of d(v) which is dominated by n-1. Hence I would conclude that the algorithm has a runtime complexity of O(|E|*|V|).
What am I missing here?
Keep the vertices in a balanced binary search tree ordered by w(v)/d(v). Finding the min is O(log |V|). Each time we delete an edge uv, we have to update u's key (by removing it and reinserting it into the tree with the new key), which takes time O(log |V|). Each of these steps is done at most |E| times.
So, basically I'm implementing an algorithm to calculate distances from one source node to every other node in a weighted graph, and if a node is in a negative cycle, it detects and marks that node as such.
My question regards the total time complexity of my algorithm. Assume V is number of nodes and E the number of edges.
The algorithm starts by asking E lines of input to specify the Edges of the graph and inserts it in the corresponding adjacency list. Such operation is O(E)
I apply the Bellman-Ford algorithm V-1 times to know the distances and then I apply the algorithm V-1 times once again to detect the Nodes in a negative cycle. This is 2 * O(VE) = O(VE).
I print a distances vector with the size V to display the distances and/or wether the node is in a negative cycle or not. O(V).
So I guess my total complexity would be O(VE + V + E). Now my question is: Since VE is almost always bigger than V+E (for large numbers, it's always!), can I drop the V+E in the complexity and make it simply O(VE)?
Yes, O(VE + V + E) simplifies to O(VE) given that V and E represent the number of vertices and edges in a graph. For a highly connected graph, E = O(V^2) and so in that case VE + V + E = O(V^3) = O(VE). For a sparse graph, E = O(V) (note, this is not necessarily a tight upper bound) and so VE + V + E = O(V^2) = O(VE). In all cases O(VE) is an appropriate upper bound on the complexity.
Yes, when dealing with asymptotic complexity, you always assume that V and E are very large (in theory, you study complexity by calculating limits when V and E approach infinity). Pretty much the same why you can write n^2 + n = O(n^2), in your case VE + V + E is O(VE).
Note that the worst-case complexity of Bellman-Ford actually is O(VE), which confirms that your reasoning is correct.
The time complexity to go over each adjacent edge of a vertex is, say, O(N), where N is number of adjacent edges. So, for V numbers of vertices the time complexity becomes O(V*N) = O(E), where E is the total number of edges in the graph. Since removing and adding a vertex from/to a queue is O(1), why is it added to the overall time complexity of BFS as O(V+E)?
I hope this is helpful to anybody having trouble understanding computational time complexity for Breadth First Search a.k.a BFS.
Queue graphTraversal.add(firstVertex);
// This while loop will run V times, where V is total number of vertices in graph.
while(graphTraversal.isEmpty == false)
currentVertex = graphTraversal.getVertex();
// This while loop will run Eaj times, where Eaj is number of adjacent edges to current vertex.
while(currentVertex.hasAdjacentVertices)
graphTraversal.add(adjacentVertex);
graphTraversal.remove(currentVertex);
Time complexity is as follows:
V * (O(1) + O(Eaj) + O(1))
V + V * Eaj + V
2V + E(total number of edges in graph)
V + E
I have tried to simplify the code and complexity computation but still if you have any questions let me know.
Considering the following Graph we see how the time complexity is O(|V|+|E|) but not O(V*E).
Adjacency List
V E
v0:{v1,v2}
v1:{v3}
v2:{v3}
v3:{}
Operating How BFS Works Step by Step
Step1:
Adjacency lists:
V E
v0: {v1,v2} mark, enqueue v0
v1: {v3}
v2: {v3}
v3: {}
Step2:
Adjacency lists:
V E
v0: {v1,v2} dequeue v0;mark, enqueue v1,v2
v1: {v3}
v2: {v3}
v3: {}
Step3:
Adjacency lists:
V E
v0: {v1,v2}
v1: {v3} dequeue v1; mark,enqueue v3
v2: {v3}
v3: {}
Step4:
Adjacency lists:
V E
v0: {v1,v2}
v1: {v3}
v2: {v3} dequeue v2, check its adjacency list (v3 already marked)
v3: {}
Step5:
Adjacency lists:
V E
v0: {v1,v2}
v1: {v3}
v2: {v3}
v3: {} dequeue v3; check its adjacency list
Step6:
Adjacency lists:
V E
v0: {v1,v2} |E0|=2
v1: {v3} |E1|=1
v2: {v3} |E2|=1
v3: {} |E3|=0
Total number of steps:
|V| + |E0| + |E1| + |E2| +|E3| == |V|+|E|
4 + 2 + 1 + 1 + 0 == 4 + 4
8 == 8
Assume an adjacency list representation, V is the number of vertices, E the number of edges.
Each vertex is enqueued and dequeued at most once.
Scanning for all adjacent vertices takes O(|E|) time, since sum of lengths of adjacency lists is |E|.
Hence The Time Complexity of BFS Gives a O(|V|+|E|) time complexity.
The other answers here do a great job showing how BFS runs and how to analyze it. I wanted to revisit your original mathematical analysis to show where, specifically, your reasoning gives you a lower estimate than the true value.
Your analysis goes like this:
Let N be the average number of edges incident to each node (N = E / V).
Each node, therefore, spends O(N) time doing operations on the queue.
Since there are V nodes, the total runtime is the O(V) · O(N) = O(V) · O(E / V) = O(E).
You are very close to having the right estimate here. The question is where the missing V term comes from. The issue here is that, weirdly enough, you can't say that O(V) · O(E / V) = O(E).
You are totally correct that the average work per node is O(E / V). That means that the total work done asympotically is bounded from above by some multiple of E / V. If we think about what BFS is actually doing, the work done per node probably looks more like c1 + c2E / V, since there's some baseline amount of work done per node (setting up loops, checking basic conditions, etc.), which is what's accounted for by the c1 term, plus some amount of work proportional to the number of edges visited (E / V, times the work done per edge). If we multiply this by V, we get that
V · (c1 + c2E / V)
= c1V + c2E
= Θ(V + E)
What's happening here is that those lovely lower-order terms that big-O so conveniently lets us ignore are actually important here, so we can't easily discard them. So that's mathematically at least what's going on.
What's actually happening here is that no matter how many edges there are in the graph, there's some baseline amount of work you have to do for each node independently of those edges. That's the setup to do things like run the core if statements, set up local variables, etc.
Performing an O(1) operation L times, results to O(L) complexity.
Thus, removing and adding a vertex from/to the Queue is O(1), but when you do that for V vertices, you get O(V) complexity.
Therefore, O(V) + O(E) = O(V+E)
Will the time complexity of BFS be not O(V) considering we only have to traverse the vertices in the adjacency list? Am I missing something here?
For the below graph represented using adjacency list for ex:
0 ->1->2->null
1->3->4->null
3->null
4->null
While creating the graph we have the adjacency list which is an array of linked lists. So my understanding is during traversal this array is available to us and it's enough if we only traverse all the elements of this array and mark each element as visited to not visit it twice. What am I missing here?
I would just like to add to above answers that if we are using an adjacency matrix instead of a adjacency list, the time complexity will be O(V^2), as we will have to go through a complete row for each vertex to check which nodes are adjacent.
You are saying that total complexity should be O(V*N)=O(E). Suppose there is no edge between any pair of vertices i.e. Adj[v] is empty for all vertex v. Will BFS take a constant time in this case? Answer is no. It will take O(V) time(more accurately θ(V)). Even if Adj[v] is empty, running the line where you check Adj[v] will itself take some constant time for each vertex. So running time of BFS is O(V+E) which means O(max(V,E)).
One of the ways that I grasped the intuition of the time complexity
O ( V + E)
is that when we traverse the graph (let's take BFS pseudocode in Java):
for(v:V){ // segment 1
if(!v.isVisited) {
q = new Queue<>();
q.add(v);
v.isVisited = true
while(!q.isEmpty) {
curr = q.poll()
for(u: curr.adjacencyList ){ //Segment 2
//do some processing
u.isVisited = true
}
}
}
}
As, we can see there are two important segments 1 and 2 which determines the time complexity.
Case 1: Consider a graph with only vertices and a few edges, sparsely connected graph (100 vertices and 2 edges).
In that case, the segment 1 would dominate the course of traversal.
Hence making, O(V) as the time complexity as segment 1 checks all vertices in graph space once.
Therefore, T.C. = O(V) (since E is negligible).
Case 2: Consider a graph with few vertices but a complete graph (6 vertices and 15 edges) (n C 2).
Here the segment 2 will dominate as the number of edges are more and the segment 2 gets evaluated 2|E| times for an undirected graph.
T.C. of first vertex processing would be,
O(1) * O(2|E|) = O(E)
The rest of the vertex will not be evaluated for the segment 1 and would just add V-1 times of processing (since they are already visited in segment 2 which is O(V).
Thus, in this case its better to say T.C. = O(E) + O(V)
So, in the worst/best case of number of edges, we have
TC(taversing) O(E) + O(V) or
= O(E+V)
The basic algorithm for BFS:
set start vertex to visited
load it into queue
while queue not empty
for each edge incident to vertex
if its not visited
load into queue
mark vertex
So I would think the time complexity would be:
v1 + (incident edges) + v2 + (incident edges) + .... + vn + (incident edges)
where v is vertex 1 to n
Firstly, is what I've said correct? Secondly, how is this O(N + E), and intuition as to why would be really nice. Thanks
Your sum
v1 + (incident edges) + v2 + (incident edges) + .... + vn + (incident edges)
can be rewritten as
(v1 + v2 + ... + vn) + [(incident_edges v1) + (incident_edges v2) + ... + (incident_edges vn)]
and the first group is O(N) while the other is O(E).
DFS(analysis):
Setting/getting a vertex/edge label takes O(1) time
Each vertex is labeled twice
once as UNEXPLORED
once as VISITED
Each edge is labeled twice
once as UNEXPLORED
once as DISCOVERY or BACK
Method incidentEdges is called once for each vertex
DFS runs in O(n + m) time provided the graph is represented by the adjacency list structure
Recall that Σv deg(v) = 2m
BFS(analysis):
Setting/getting a vertex/edge label takes O(1) time
Each vertex is labeled twice
once as UNEXPLORED
once as VISITED
Each edge is labeled twice
once as UNEXPLORED
once as DISCOVERY or CROSS
Each vertex is inserted once into a sequence Li
Method incidentEdges is called once for each vertex
BFS runs in O(n + m) time provided the graph is represented by the adjacency list structure
Recall that Σv deg(v) = 2m
Very simplified without much formality: every edge is considered exactly twice, and every node is processed exactly once, so the complexity has to be a constant multiple of the number of edges as well as the number of vertices.
An intuitive explanation to this is by simply analysing a single loop:
visit a vertex -> O(1)
a for loop on all the incident edges -> O(e) where e is a number of edges incident on a given vertex v.
So the total time for a single loop is O(1)+O(e). Now sum it for each vertex as each vertex is visited once. This gives
For every V
=>
O(1)
+
O(e)
=> O(V) + O(E)
Time complexity is O(E+V) instead of O(2E+V) because if the time complexity is n^2+2n+7 then it is written as O(n^2).
Hence, O(2E+V) is written as O(E+V)
because difference between n^2 and n matters but not between n and 2n.
I think every edge has been considered twice and every node has been visited once, so the total time complexity should be O(2E+V).
Short but simple explanation:
I the worst case you would need to visit all the vertex and edge hence
the time complexity in the worst case is O(V+E)
In Bfs, each neighboring vertex is inserted once into a queue. This is done by looking at the edges of the vertex. Each visited vertex is marked so it cannot be visited again: each vertex is visited exactly once, and all edges of each vertex are checked. So the complexity of BFS is V+E.
In DFS, each node maintains a list of all its adjacent edges, then, for each node, you need to discover all its neighbors by traversing its adjacency list just once in linear time. For a directed graph, the sum of the sizes of the adjacency lists of all the nodes is E(total number of edges). So, the complexity of DFS is O(V + E).
It's O(V+E) because each visit to v of V must visit each e of E where |e| <= V-1. Since there are V visits to v of V then that is O(V). Now you have to add V * |e| = E => O(E). So total time complexity is O(V + E).