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Ternary Search

Ternary Search. Concept, ICPC 4504. Convex Function and Concave Function. For any two points x 1 and x 2 and any t ∈ [0, 1 ] A function is concave if − f is convex. f ( tx 1 + (1 − t ) x 2 ) ≤ tf ( x 1 ) + (1 − t ) f ( x 2 ). f ( x ). f ( x 2 ). tf ( x 1 ) + (1 − t ) f ( x 2 ).

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Ternary Search

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  1. Ternary Search Concept, ICPC 4504

  2. Convex Function and Concave Function • For any two points x1 and x2 and any t ∈ [0, 1] • A function is concave if −f is convex f(tx1 + (1 − t)x2) ≤ tf(x1) + (1 − t)f(x2) f(x) f(x2) tf(x1) + (1 − t)f(x2) f(x1) f(tx1 + (1 − t)x2) x1 x2 tx1 + (1 − t)x2

  3. Examples f(x) = ax + b f(x) = x2 f(x) = |x|

  4. Finding the Minimum • 給予convex function f，並已知f在(low, up) 之間出現最小值，求值為何 • 像在具有單調性的函數上binary search 的方式，不斷丟棄不可能存在答案的區間來逼進答案 • ternary search 也是divide and conquer binary search ternary search f(mid) f(ml) t f(mr) low mid up low ml mr up f(mid) > t (目標的函數值) 答案不會在mid和up之間 f(ml) >f(mr) 最小值不會在low和ml之間

  5. Algorithm • 假設我們已經知道minimum 所在的區間為(l, r)，任取兩點ml, mr，並且l < ml < mr < r，討論三種情況 • 不管mr切的位置在minimum 的哪一側，如果f(ml) > f(mr)，答案一定不在 low和ml之間 (1) f(ml) > f(mr) f(ml) f(ml) f(mr) f(mr) ml mr ml mr

  6. Algorithm • 同樣地，如果f(ml) < f(mr)，那麼mr到up之間這段就可以丟掉 (2) f(ml) < f(mr) f(mr) f(mr) f(ml) f(ml) ml mr ml mr

  7. Algorithm • 如果函數值相等，minimum 的範圍就可以縮小到ml到mr之間 • 這個case 可以和前面兩個任一合併，即在相等的時候選擇只丟掉左側或右側 (3) f(ml) = f(mr) f(ml) = f(mr) ml mr

  8. Examples of Implementation (I) doubleternary_search() { double low =kLeftBound, up =kRightBound, ml,mr; while(low +kEps< up) { ml = (low + up) /2,mr= (ml + up) /2; // or ml = (2 * low + up) / 3, mr= (low + 2 * up) / 3; if(f(ml) < f(mr)) up =mr; else low = ml; } return low; }

  9. Examples of Implementation (I) • 使用範圍大小來當迴圈的終止條件的缺點是，可能會因為浮點數的計算誤差陷入無窮迴圈 • 可以估計出大概切幾次之後範圍能縮小到能接受的大小，將迴圈改成固定的次數 • 每次縮小為1/2或3/4，以(3/4)k來推算迴圈的次數 • 每次縮小為2/3 ml = (low + up) /2,mr= (ml + up) /2; ml = (2* low + up) /3,mr= (low +2* up) /3;

  10. Examples of Implementation (II) doubleternary_search() { double low =kLeftBound, up =kRightBound, ml,mr; for(inti=0;i<kNumIterations; ++i) { ml = (low + up) /2,mr= (ml + up) /2; if(f(ml) < f(mr)) up =mr; else low = ml; } return low; }

  11. Properties • 兩個convex function 的sum 仍是一個convex function[?] • 兩個 convex function 的 max 仍是一個 convex function[?]

  12. ICPC 4504 • 平面上給定n個點(記為pi, 1 ≤ i ≤ n)，求在x軸上找一點p，使最小，d為兩點在平面上的距離 f(p) = max{ d(p, pi) } f(p = p1) p1

  13. ICPC 4504 • 平面上給定n個點(記為pi, 1 ≤ i ≤ n)，求在x軸上找一點p，使最小，d為兩點在平面上的距離 f(p) = max{ d(p, pi) } f(p = p2) p2

  14. Observation • 觀察當p在x軸上往正向移動時，其x座標對p到其中某一個給定點(xi, yi) 的函數圖形 • 不把根號開掉，實際上就是一個二次函數 yi xi (xi, yi)

  15. Observation • 有了各個點畫出來的函數fi(x)，我們最後取的是F(x) = max{ fi(x) } • 因為每個 fi(x) 都是 convex function，而且 F(x) 是所有函數的 max，故 F(x) 也是一個 convex function F(x) yi yj xi xj (xi, yi) (xj, yj)

  16. Idea • 尋找以p點在原本座標系的x座標對最遠點距離所繪出的函數圖形，發生最小值時的x座標為何 • 選擇合理的起始範圍，並從這組左右界開始以ternary search 來逼進答案

  17. Appendix • 兩個convex function 的sum 仍是一個convex function 令 h(x) = f(x) + g(x)，其中 f(x) 和 g(x) 是 convex functions 任取兩點 x1和 x2以及 t ∈ [0, 1] h(tx1 + (1 − t)x2) = f(tx1 + (1 − t)x2) + g(tx1 + (1 − t)x2) ≤ tf(x1) + (1 − t)f(x2) + tg(x1) + (1 − t)g(x2) = t[f(x1) + g(x1)] + (1 − t)[f(x2) + g(x2)] = th(x1) + (1 − t)h(x2)

  18. Appendix • 兩個 convex function 的 max 仍是一個 convex function 令 h(x) = max{ f(x), g(x) }，其中 f(x) 和 g(x) 是 convex functions 任取兩點 x1和 x2以及 t ∈ [0, 1] f(tx1+ (1 − t)x2) ≤ tf(x1) + (1 − t)f(x2) g(tx1+ (1 − t)x2) ≤ tg(x1) + (1 − t)g(x2) tf(x1) + (1 − t)f(x2) ≤ th(x1) + (1 − t)h(x2) tg(x1) + (1 − t)g(x2) ≤ th(x1) + (1 − t)h(x2) h(tx1 + (1 − t)x2) = max{f(tx1 + (1 − t)x2), g(tx1 + (1 − t)x2) } ≤ max{tf(x1) + (1 − t)f(x2), tg(x1) + (1 − t)g(x2) } ≤ th(x1) + (1 − t)h(x2)

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