Extra Algorithms added

Graphs/ArticulationPoints.py

@@ -0,0 +1,44 @@
+# Finding Articulation Points in Undirected Graph
+def computeAP(l):
+    n = len(l)
+    outEdgeCount = 0
+    low = [0] * n
+    visited = [False] * n
+    isArt = [False] * n
+
+    def dfs(root, at, parent, outEdgeCount):
+        if parent == root:
+            outEdgeCount += 1
+        visited[at] = True
+        low[at] = at
+
+        for to in l[at]:
+            if to == parent:
+                pass
+            elif not visited[to]:
+                outEdgeCount = dfs(root, to, at, outEdgeCount)
+                low[at] = min(low[at], low[to])
+
+                # AP found via bridge
+                if at < low[to]:
+                    isArt[at] = True
+                # AP found via cycle
+                if at == low[to]:
+                    isArt[at] = True
+            else:
+                low[at] = min(low[at], to)
+        return outEdgeCount
+
+    for i in range(n):
+        if not visited[i]:
+            outEdgeCount = 0
+            outEdgeCount = dfs(i, i, -1, outEdgeCount)
+            isArt[i] = (outEdgeCount > 1)
+
+    for x in range(len(isArt)):
+        if isArt[x] == True:
+            print(x, end=" ")
+
+# Adjacency list of graph
+l = {0:[1,2], 1:[0,2], 2:[0,1,3,5], 3:[2,4], 4:[3], 5:[2,6,8], 6:[5,7], 7:[6,8], 8:[5,7]}
+computeAP(l)

Graphs/CheckBipartiteGraph_BFS.py

@@ -0,0 +1,43 @@
+# Check whether Graph is Bipartite or Not using BFS
+
+# A Bipartite Graph is a graph whose vertices can be divided into two independent sets,
+# U and V such that every edge (u, v) either connects a vertex from U to V or a vertex
+# from V to U. In other words, for every edge (u, v), either u belongs to U and v to V,
+# or u belongs to V and v to U. We can also say that there is no edge that connects
+# vertices of same set.
+def checkBipartite(l):
+    queue = []
+    visited = [False] * len(l)
+    color = [-1] * len(l)
+
+    def bfs():
+        while(queue):
+            u = queue.pop(0)
+            visited[u] = True
+
+            for neighbour in l[u]:
+
+                if neighbour == u:
+                    return False
+
+                if color[neighbour] == -1:
+                    color[neighbour] = 1 - color[u]
+                    queue.append(neighbour)
+
+                elif color[neighbour] == color[u]:
+                    return False
+
+        return True
+
+    for i in range(len(l)):
+        if not visited[i]:
+            queue.append(i)
+            color[i] = 0
+            if bfs() == False:
+                return False
+
+    return True
+
+# Adjacency List of graph
+l = {0:[1,3], 1:[0,2], 2:[1,3], 3:[0,2]}
+print(checkBipartite(l))

Graphs/FindingBridges.py

@@ -0,0 +1,31 @@
+# Finding Bridges in Undirected Graph
+def computeBridges(l):
+    id = 0
+    n = len(l) # No of vertices in graph
+    low = [0] * n
+    visited = [False] * n
+
+    def dfs(at, parent, bridges, id):
+        visited[at] = True
+        low[at] = id
+        id += 1
+        for to in l[at]:
+            if to == parent:
+                pass
+            elif not visited[to]:
+                dfs(to, at, bridges, id)
+                low[at] = min(low[at], low[to])
+                if at < low[to]:
+                    bridges.append([at, to])
+            else:
+                # This edge is a back edge and cannot be a bridge
+                low[at] = min(low[at], to)
+
+    bridges = []
+    for i in range(n):
+        if (not visited[i]):
+            dfs(i, -1, bridges, id)
+    print(bridges)
+            
+l = {0:[1,2], 1:[0,2], 2:[0,1,3,5], 3:[2,4], 4:[3], 5:[2,6,8], 6:[5,7], 7:[6,8], 8:[5,7]}
+computeBridges(l)

Graphs/KahnsAlgorithm_long.py

@@ -0,0 +1,30 @@
+# Finding longest distance in Directed Acyclic Graph using KahnsAlgorithm
+def longestDistance(l):
+    indegree = [0] * len(l)
+    queue = []
+    longDist = [1] * len(l)
+
+    for key, values in l.items():
+        for i in values:
+            indegree[i] += 1
+
+    for i in range(len(indegree)):
+        if indegree[i] == 0:
+            queue.append(i)
+
+    while(queue):
+        vertex = queue.pop(0)
+        for x in l[vertex]:
+            indegree[x] -= 1
+
+            if longDist[vertex] + 1 > longDist[x]:
+                longDist[x] =  longDist[vertex] + 1
+
+            if indegree[x] == 0:
+                queue.append(x)
+
+    print(max(longDist))
+
+# Adjacency list of Graph
+l = {0:[2,3,4], 1:[2,7], 2:[5], 3:[5,7], 4:[7], 5:[6], 6:[7], 7:[]}
+longestDistance(l)

Graphs/KahnsAlgorithm_topo.py

@@ -0,0 +1,32 @@
+# Kahn's Algorithm is used to find Topological ordering of Directed Acyclic Graph using BFS
+def topologicalSort(l):
+    indegree = [0] * len(l)
+    queue = []
+    topo = []
+    cnt = 0
+
+    for key, values in l.items():
+        for i in values:
+            indegree[i] += 1
+
+    for i in range(len(indegree)):
+        if indegree[i] == 0:
+            queue.append(i)
+
+    while(queue):
+        vertex = queue.pop(0)
+        cnt += 1
+        topo.append(vertex)
+        for x in l[vertex]:
+            indegree[x] -= 1
+            if indegree[x] == 0:
+                queue.append(x)
+
+    if cnt != len(l):
+        print("Cycle exists")
+    else:
+        print(topo)
+
+# Adjacency List of Graph
+l = {0:[1,2], 1:[3], 2:[3], 3:[4,5], 4:[], 5:[]}
+topologicalSort(l)

Graphs/MinimumSpanningTree_Prims.py

@@ -0,0 +1,111 @@
+import sys
+from collections import defaultdict
+
+def PrimsAlgorithm(l):
+
+    nodePosition = []
+    def getPosition(vertex):
+        return nodePosition[vertex]
+
+    def setPosition(vertex, pos):
+        nodePosition[vertex] = pos
+
+    def topToBottom(heap, start, size, positions):
+        if start > size // 2 - 1:
+            return
+        else:
+            if 2 * start + 2 >= size:
+                m = 2 * start + 1
+            else:
+                if heap[2 * start + 1] < heap[2 * start + 2]:
+                    m = 2 * start + 1
+                else:
+                    m = 2 * start + 2
+            if heap[m] < heap[start]:
+                temp, temp1 = heap[m], positions[m]
+                heap[m], positions[m] = heap[start], positions[start]
+                heap[start], positions[start] = temp, temp1
+
+                temp = getPosition(positions[m])
+                setPosition(positions[m], getPosition(positions[start]))
+                setPosition(positions[start], temp)
+
+                topToBottom(heap, m, size, positions)
+
+    # Update function if value of any node in min-heap decreases
+    def bottomToTop(val, index, heap, position):
+        temp = position[index]
+
+        while(index != 0):
+            if index % 2 == 0:
+                parent = int( (index-2) / 2 )
+            else:
+                parent = int( (index-1) / 2 )
+
+            if val < heap[parent]:
+                heap[index] = heap[parent]
+                position[index] = position[parent]
+                setPosition(position[parent], index)
+            else:
+                heap[index] = val
+                position[index] = temp
+                setPosition(temp, index)
+                break
+            index = parent
+        else:
+            heap[0] = val
+            position[0] = temp
+            setPosition(temp, 0)
+
+    def heapify(heap, positions):
+        start = len(heap) // 2 - 1
+        for i in range(start, -1, -1):
+            topToBottom(heap, i, len(heap), positions)
+
+    def deleteMinimum(heap, positions):
+        temp = positions[0]
+        heap[0] = sys.maxsize
+        topToBottom(heap, 0, len(heap), positions)
+        return temp
+
+    visited = [0 for i in range(len(l))]
+    Nbr_TV  = [-1 for i in range(len(l))] # Neighboring Tree Vertex of selected vertex
+    # Minimum Distance of explored vertex with neighboring vertex of partial tree formed in graph
+    Distance_TV = [] # Heap of Distance of vertices from their neighboring vertex
+    Positions = []
+
+    for x in range(len(l)):
+        p = sys.maxsize
+        Distance_TV.append(p)
+        Positions.append(x)
+        nodePosition.append(x)
+
+    TreeEdges = []
+    visited[0] = 1
+    Distance_TV[0] = sys.maxsize
+    for x in l[0]:
+        Nbr_TV[ x[0] ] = 0
+        Distance_TV[ x[0] ] = x[1]
+    heapify(Distance_TV, Positions)
+
+    for i in range(1, len(l)):
+        vertex = deleteMinimum(Distance_TV, Positions)
+        if visited[vertex] == 0:
+            TreeEdges.append((Nbr_TV[vertex], vertex))
+            visited[vertex] = 1
+            for v in l[vertex]:
+                if visited[v[0]] == 0 and v[1] < Distance_TV[ getPosition(v[0]) ]:
+                    Distance_TV[ getPosition(v[0]) ] = v[1]
+                    bottomToTop(v[1], getPosition(v[0]), Distance_TV, Positions)
+                    Nbr_TV[ v[0] ] = vertex
+    return TreeEdges
+
+# < --------- Prims Algorithm --------- >
+n = int(input("Enter number of vertices: "))
+e = int(input("Enter number of edges: "))
+adjlist = defaultdict(list)
+for x in range(e):
+    l = [int(x) for x in input().split()]
+    adjlist[l[0]].append([ l[1], l[2] ])
+    adjlist[l[1]].append([ l[0], l[2] ])
+print(PrimsAlgorithm(adjlist))

Maths/BasicMaths.py

@@ -0,0 +1,70 @@
+import math
+
+def primeFactors(n):
+    pf = []
+    while n % 2 == 0:
+        pf.append(2)
+        n = int(n / 2)
+        
+    for i in range(3, int(math.sqrt(n))+1, 2):
+        while n % i == 0:
+            pf.append(i)
+            n = int(n / i)
+    
+    if n > 2:
+        pf.append(n)
+    
+    return pf
+
+def numberOfDivisors(n):
+    div = 1
+    
+    temp = 1
+    while n % 2 == 0:
+        temp += 1
+        n = int(n / 2)
+    div = div * (temp) 
+    
+    for i in range(3, int(math.sqrt(n))+1, 2):
+        temp = 1
+        while n % i == 0:
+            temp += 1
+            n = int(n / i)
+        div = div * (temp)
+    
+    return div
+
+def sumOfDivisors(n):
+    s = 1
+    
+    temp = 1
+    while n % 2 == 0:
+        temp += 1
+        n = int(n / 2)
+    if temp > 1:
+        s *= (2**temp - 1) / (2 - 1) 
+    
+    for i in range(3, int(math.sqrt(n))+1, 2):
+        temp = 1
+        while n % i == 0:
+            temp += 1
+            n = int(n / i)
+        if temp > 1:
+            s *= (i**temp - 1) / (i - 1)
+    
+    return s
+
+def eulerPhi(n):
+    l = primeFactors(n)
+    l = set(l)
+    s = n
+    for x in l:
+        s *= (x - 1)/x 
+    return s   
+
+print(primeFactors(100))
+print(numberOfDivisors(100))
+print(sumOfDivisors(100))
+print(eulerPhi(100))
+    
+    

Maths/SegmentedSieve.py

@@ -0,0 +1,46 @@
+import math
+
+def sieve(n):
+    in_prime = []
+    start = 2
+    end   = int(math.sqrt(n)) # Size of every segment
+    temp = [True] * (end + 1)
+    prime = []
+    
+    while(start <= end):
+        if temp[start] == True:
+            in_prime.append(start)
+            for i in range(start*start, end+1, start):
+                if temp[i] == True:
+                    temp[i] = False
+        start += 1
+    prime += in_prime
+    
+    low = end + 1
+    high = low + end - 1
+    if high > n:
+        high = n
+    
+    while(low <= n):
+        temp = [True] * (high-low+1)
+        for each in in_prime:
+            
+            t = math.floor(low / each) * each
+            if t < low:
+                t += each
+            
+            for j in range(t, high+1, each):
+                temp[j - low] = False
+                
+        for j in range(len(temp)):
+            if temp[j] == True:
+                prime.append(j+low)
+        
+        low = high + 1
+        high = low + end - 1
+        if high > n:
+            high = n
+            
+    return prime
+
+print(sieve(10**6))

Maths/SieveOfEratosthenes.py

@@ -0,0 +1,24 @@
+import math
+n = int(input("Enter n: "))
+
+def sieve(n):
+    l = [True] * (n+1)
+    prime = []
+    start = 2
+    end   = int(math.sqrt(n))
+    while(start <= end):
+        if l[start] == True:
+            prime.append(start)
+            for i in range(start*start, n+1, start):
+                if l[i] == True:
+                    l[i] = False
+        start += 1
+    
+    for j in range(end+1,n+1):
+        if l[j] == True:
+            prime.append(j)
+    
+    return prime
+
+print(sieve(n))
+