Create number-of-ways-to-reach-a-position-after-exactly-k-steps.cpp

Author: kamyu104

C++/number-of-ways-to-reach-a-position-after-exactly-k-steps.cpp

@@ -0,0 +1,51 @@
+// Time:  O(k)
+// Space: O(k)
+
+// combinatorics
+class Solution {
+public:
+    int numberOfWays(int startPos, int endPos, int k) {
+        const int r = k - abs(endPos - startPos);
+        return r >= 0 && r % 2 == 0 ? nCr(k, r / 2) : 0;
+    }
+ 
+private:
+    int nCr(int n, int k) {
+        while (size(inv_) <= n) {  // lazy initialization
+            fact_.emplace_back(mulmod(fact_.back(), size(inv_)));
+            inv_.emplace_back(mulmod(inv_[MOD % size(inv_)], MOD - MOD / size(inv_)));  // https://cp-algorithms.com/algebra/module-inverse.html
+            inv_fact_.emplace_back(mulmod(inv_fact_.back(), inv_.back()));
+        }
+        return mulmod(mulmod(fact_[n], inv_fact_[n - k]), inv_fact_[k]);
+    }
+
+    uint32_t addmod(uint32_t a, uint32_t b) {  // avoid overflow
+        a %= MOD, b %= MOD;
+        if (MOD - a <= b) {
+            b -= MOD;  // relied on unsigned integer overflow in order to give the expected results
+        }
+        return a + b;
+    }
+
+    // reference: https://stackoverflow.com/questions/12168348/ways-to-do-modulo-multiplication-with-primitive-types
+    uint32_t mulmod(uint32_t a, uint32_t b)  {  // avoid overflow
+        a %= MOD, b %= MOD;
+        uint32_t result = 0;
+        if (a < b) {
+            swap(a, b);
+        }
+        while (b > 0)  { 
+            if (b % 2 == 1) {
+                result = addmod(result, a);
+            }
+            a = addmod(a, a);
+            b /= 2; 
+        } 
+        return result; 
+    }
+   
+    static const uint32_t MOD = 1e9 + 7;
+    vector<int> fact_ = {1, 1};
+    vector<int> inv_ = {1, 1};
+    vector<int> inv_fact_ = {1, 1};
+};