Concepts158
Divide and Conquer DP Optimization
Divide and Conquer DP optimization speeds up DP transitions of the form dp[i][j] = min over k of dp[i-1][k] + C(k, j) when the optimal k is monotone in j.
Convex Hull Trick - Dynamic (LineContainer)
Dynamic Convex Hull Trick (LineContainer) maintains the lower or upper envelope of lines y = m x + b with O(log n) insertion and O(log n) query for arbitrary insertion order.
Convex Hull Trick (CHT)
The Convex Hull Trick (CHT) speeds up dynamic programs where each state is a minimum over linear functions, such as dp[i] = min_j (dp[j] + b[j] × a[i]).
Knuth Optimization
Knuth Optimization speeds up a class of interval dynamic programming (DP) from O(n^3) to O(n^2) by exploiting the monotonicity of optimal split points.
Interval DP
Interval DP solves problems where the optimal answer for a segment [i, j] depends on answers of its subsegments.
Tree DP - Matching and Covering
Tree DP solves matching, vertex cover, and independent set on trees in linear time using small state transitions per node.
Bitmask DP
Bitmask DP compresses the state of a subset of n elements into an integer mask, enabling elegant dynamic programming over all subsets.
Longest Common Subsequence
The Longest Common Subsequence (LCS) between two sequences is the longest sequence that appears in both, not necessarily contiguously.
Bitmask DP - Subset Enumeration
Bitmask DP subset enumeration lets you iterate all submasks of a given mask using the idiom for (s = mask; s > 0; s = (s - 1) & mask).
DP on Trees
DP on trees is a technique that computes answers for each node by combining results from its children using a post-order DFS.
Edit Distance
Edit distance (Levenshtein distance) measures the minimum number of inserts, deletes, and replaces needed to turn one string into another.
2-SAT
2-SAT solves Boolean formulas where every clause has exactly two literals, and it is solvable in linear time relative to the size of the implication graph.