my solution for 1514

Author: jeremymanning

problems/1514/jeremymanning.md

@@ -1,11 +1,93 @@
 # [Problem 1514: Path with Maximum Probability](https://leetcode.com/problems/path-with-maximum-probability/description/?envType=daily-question)
 
 ## Initial thoughts (stream-of-consciousness)
+- First we need a way to represent the graph more conveniently.  I'm thinking we should have a hash table whose keys are nodes and whose values are lists of connected nodes + probabilities.  E.g., for the first given example, `graph[0] = [(2, 0.2), (1, 0.5)]`, etc.  We can build up this hash table with a single pass through the edge list + success probability list:
+```python
+graph = {}
+for edge, prob in zip(edges, succProb):
+    if edge[0] in graph:
+        graph[edge[0]].append((edge[1], prob))
+    else:
+        graph[edge[0]] = [(edge[1], prob)]
+
+    if edge[1] in graph:
+        graph[edge[1]].append((edge[0], prob))
+    else:
+        graph[edge[1]] = [(edge[0], prob)]
+```
+- Now I think we can build up paths using BFS or DFS starting from `start_node`:
+    - If we hit a loop, stop and delete that path from the queue/stack
+    - If we reach `end_node`:
+        - Go back and compute the probability of this path
+        - If it's higher than the probability of the current best path (initialized to `0`), replace the best probability
+        - Then delete the path from the queue/stack
+- Potential pitfalls:
+    - We might run out of memory if we build up the paths one node at a time.
+    - Ideally we could instead just build up something like probability of the path so far + last node reached.  But then I'm not sure we'll be able to detect loops...
+    - The probabilities could get very small, leading to rounding errors.  We could instead track *log* probabilities (summing as we go) and then exponentiate at the end.
+- Instead of BFS/DFS, maybe we want to use a heap to keep the paths sorted by max probability.
 
 ## Refining the problem, round 2 thoughts
+We could use the built-in `heapq` to do this (according to the documentation `heapq` is a max heap, but we can just multiply the probabilities by -1 so that the most probable paths appear at the "top" of the heap):
+    - First push the start of the path (`(-1.0, start_node)`) onto the heap
+    - Keep track of `n` probabilities (of reaching each node).  Initialize to 0.0, except the start node (initialize to 1.0).
+    - While the heap is not empty:
+        - Pop the most probable path `(prob, last_node)`
+        - If we're at the end node, return `prob`
+        - Otherwise:
+            - For each neighbor `x` of `last_node` (connected with probability `new_prob`):
+                - Probability of path ending at `x` is `-prob * new_prob`
+                - If `-prob * new_prob` is greater than `probabilities[x]`:
+                    - Update `probabilities[x]` -- this ensures we only push more probable paths than we've already found onto the heap.  This also avoids loops, since visiting an already-visited node will necessarily result in a lower probability path to that node.
+                    - Push `(prob * new_prob, x)` on the heap
+    - If the heap empties before we found the end, just return 0.0.
 
 ## Attempted solution(s)
 ```python
-class Solution:  # paste your code here!
-    ...
+import heapq
+
+class Solution:
+    def maxProbability(self, n, edges, succProb, start, end):
+        # build the graph
+        graph = {}
+        for x in range(n):
+            graph[x] = []
+        for edge, prob in zip(edges, succProb):
+            graph[edge[0]].append((edge[1], prob))
+            graph[edge[1]].append((edge[0], prob))
+            
+        # max heap
+        heap = [(-1.0, start)]
+        probabilities = [0.0] * n
+        probabilities[start] = 1.0
+
+        while heap:
+            p, node = heapq.heappop(heap)
+            if node == end:
+                return -p
+
+            for x, xp in graph[node]:
+                new_prob = p * xp   # remember: this is negative
+                if -new_prob > probabilities[x]:
+                    probabilities[x] = -new_prob
+                    heapq.heappush(heap, (new_prob, x))
+
+        # end not found :(
+        return 0.0
+```
+- Given test cases pass
+- Another example:
+```python
+n = 7
+edges = [[0,1],[1,2],[2,3],[3,4],[4,5],[5,6]]
+succProb = [0.01, 0.01, 0.01, 0.01, 0.01, 0.1]
+start_node = 0
+end_node = 6
 ```
+- passes!
+- Ok, submitting!
+
+![Screenshot 2024-08-27 at 12 07 09 AM](https://github.com/user-attachments/assets/3d64e697-53a5-4cbf-a3b4-d397ea27a0f5)
+
+Solved!
+