diff --git a/divide-and-conquer/notes.md b/divide-and-conquer/notes.md index 95cc7aaa36a269c8055d9d5c3401bd877b4b7add..7ba5144320e67a958209ba92075589c5011307ad 100644 --- a/divide-and-conquer/notes.md +++ b/divide-and-conquer/notes.md @@ -104,24 +104,39 @@ Problem: Find the median of an odd-length list in average-case Θ(n) time. Use ``` In: [8, 7, 6, 9, 5], k = 2 - In: […], k = 2 - In: […], k = 2 - Split: […], …, […] - Out: … + * + i + j + In: [8, 7, 6, 5, 9], k = 2 + * + i + j (crossed) + i + In: [5, 7, 6, 8, 9], k = 2 + Split: [5, 7, 6], 8, [9] + Out: 7 | | | - In: […], k = … - In: […], k = … - Split: […], …, […] - Out: … + In: [5, 7, 6], k = 2 + * + i + j (crossed) + i + In: [5, 7, 6], k = 2 + Split: [], 5, [7, 6] + Out: 7 | | | - In: […], k = … - In: […], k = … - Split: […], …, […] - Out: … + In: [7, 6], k = 1 + * + i + j (crossed) + i + In: [6, 7], k = 1 + Split: [6], 7, [] + Out: 7 ``` ## Analysis @@ -129,9 +144,13 @@ Problem: Find the median of an odd-length list in average-case Θ(n) time. Use * Uneven split: T(n) = T(n - 1) + Θ(n) for n ≫ 0 - T(n) = Θ(⋯) + T(n) = Θ(n²) * Even split: T(n) ≈ T(n/2) + Θ(n) for n ≫ 0 - T(n) = Θ(⋯) + T(n) = "about" Θ(1) + Θ(n) = Θ(n) + +* Land on the pivot: + + T(n) = 0 + Θ(n) = Θ(n) diff --git a/divide-and-conquer/src/features/median/kthSmallest.js b/divide-and-conquer/src/features/median/kthSmallest.js index 7125d1f2a0d5e3053d9de9c59242e5be9e6b6f52..26e87bb8b538d372c873512ce263560fbd8921ad 100644 --- a/divide-and-conquer/src/features/median/kthSmallest.js +++ b/divide-and-conquer/src/features/median/kthSmallest.js @@ -31,5 +31,15 @@ export function kthSmallest(sequence, k) { k >= 0 && k < sequence.length, 'Tried to find the kth smallest element when k is out of bounds.', ); - return sequence[0]; // TODO: stub + if (sequence.length < 2) { + return sequence[0]; + } + const [left, pivot, right] = partition(sequence); + if (k === left.length) { + return pivot; + } + if (k < left.length) { + return kthSmallest(left, k); + } + return kthSmallest(right, k - left.length - 1); }