#167 is completed with a DP technique + buttom up approach

Author: Hamid Gasmi

6-dynamic-programming-applications-in-machine-learning-and-genomics/1-sequence-alignment-1/longest_path_grid.py

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+import sys
+
+# 1. Express a solution mathematically: Let's be a Matrix M of (n+1) x (m+1):
+#     For 0 <= r <= n and 0 <= c <= m, M[r,c] contains the longest path from the source (0, 0) to (r, c) 
+#     M[0 , 0] = 0
+#     M[0 , c] = M[0 , c - 1] + right[0, c] for 1 <= c <= m
+#     M[r , 0] = M[r - 1 , 0] + down[r , 0] for 1 <= r <= n
+#     M[r , c] = max(M[r - 1 , c] + down[r - 1 , c], M[r , c - 1] + right[r ,  c - 1])
+# 2. Proof:
+#     Let's assume that there is a vertice (r, c) that belongs to the optimal path P with a the longest path length |P|
+#     But M[r , c] < max(M[r - 1 , c] + down[r - 1 , c], M[r , c - 1] + right[r ,  c - 1])
+#     This means that if we replace M[r , c] with max(M[r - 1 , c] + down[r - 1 , c], M[r , c - 1] + right[r ,  c - 1])
+#     The new path P length |P'| will be greater than |P| ==> contradiction with the fact that |P| was the longest path
+# 3. Implementation:
+#    Buttom up solution
+#    Running Time: O(nm) (Quadratic)
+#    Space complexity: O(nm) (Quadratic)
+class Solution:
+    def __init__(self, n, m):
+        self.rows_count = n + 1
+        self.columns_count = m + 1
+    
+    def longest_path(self, down, right):
+        
+        M = [ [0 for _ in range(self.columns_count)] for _ in range(self.rows_count) ]
+        
+        for c in range(1, self.columns_count, 1):
+            M[0][c] = M[0][c - 1] + right[0][c - 1]
+
+        for r in range(1, self.rows_count, 1):
+            M[r][0] = M[r - 1][0] + down[r - 1][0]
+        
+        for r in range(1, self.rows_count, 1):
+            for c in range(1, self.columns_count, 1):
+                candidate_predecesor_top = M[r - 1][c] + down[r - 1][c]
+                candidate_predecesor_left = M[r][c - 1] + right[r][c - 1]
+                M[r][c] = max(candidate_predecesor_top, candidate_predecesor_left)
+
+        return M[self.rows_count - 1][self.columns_count - 1]
+
+if __name__ == "__main__":
+    n,m = map(int, sys.stdin.readline().strip().split())
+    down = [list(map(int, sys.stdin.readline().strip().split()))
+            for _ in range(n)]
+    sys.stdin.readline()
+    right = [list(map(int, sys.stdin.readline().strip().split()))
+            for _ in range(n+1)]
+
+    s = Solution(n, m)
+    print(s.longest_path(down, right))

6-dynamic-programming-applications-in-machine-learning-and-genomics/1-sequence-alignment-1/longest_path_grid_test/01

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