{ "id": "searching/ternary-search", "version": 1, "deterministic": true, "url": "/dsa/searching/ternary-search", "topic": { "slug": "searching", "title": "Searching" }, "title": "Ternary Search", "tagline": "Cut into thirds instead of halves — and discover it's actually slower.", "vizKind": "array", "complexity": { "time": { "best": "O(1)", "average": "O(log n)", "worst": "O(log n)" }, "space": "O(1)", "growth": "log", "note": "Still logarithmic — but with a worse constant than binary search. Roughly 2·log₃(n) ≈ 1.26·log₂(n) comparisons, against binary search's 1.0. Same complexity class, measurably more work." }, "code": { "javascript": "function ternarySearch(a, target) {\n let lo = 0, hi = a.length - 1;\n while (lo <= hi) {\n const third = Math.floor((hi - lo) / 3);\n const m1 = lo + third;\n const m2 = hi - third;\n if (a[m1] === target) return m1;\n if (a[m2] === target) return m2;\n if (target < a[m1]) hi = m1 - 1;\n else if (target > a[m2]) lo = m2 + 1;\n else { lo = m1 + 1; hi = m2 - 1; }\n }\n return -1;\n}", "python": "def ternary_search(a, target):\n lo, hi = 0, len(a) - 1\n while lo <= hi:\n third = (hi - lo) // 3\n m1 = lo + third\n m2 = hi - third\n if a[m1] == target: return m1\n if a[m2] == target: return m2\n if target < a[m1]: hi = m1 - 1\n elif target > a[m2]: lo = m2 + 1\n else: lo, hi = m1 + 1, m2 - 1\n\n return -1\n", "java": "int ternarySearch(int[] a, int target) {\n int lo = 0, hi = a.length - 1;\n while (lo <= hi) {\n int third = (hi - lo) / 3;\n int m1 = lo + third;\n int m2 = hi - third;\n if (a[m1] == target) return m1;\n if (a[m2] == target) return m2;\n if (target < a[m1]) hi = m1 - 1;\n else if (target > a[m2]) lo = m2 + 1;\n else { lo = m1 + 1; hi = m2 - 1; }\n }\n return -1;\n}", "cpp": "int ternarySearch(vector a, int target) {\n int lo = 0, hi = a.size() - 1;\n while (lo <= hi) {\n int third = (hi - lo) / 3;\n int m1 = lo + third;\n int m2 = hi - third;\n if (a[m1] == target) return m1;\n if (a[m2] == target) return m2;\n if (target < a[m1]) hi = m1 - 1;\n else if (target > a[m2]) lo = m2 + 1;\n else { lo = m1 + 1; hi = m2 - 1; }\n }\n return -1;\n}" }, "defaultInput": [ 3, 7, 12, 18, 24, 31, 40, 55, 63, 77, 82, 91 ], "defaultTarget": 82, "inputHint": "Count the comparisons, then try the same list on Binary Search.", "pitfalls": [ { "wrong": "Assuming more splits means fewer comparisons.", "right": "Fewer rounds, but more comparisons per round. Ternary does about 26% more work than binary search. Count comparisons, not iterations." }, { "wrong": "Using it to search a sorted array in real code.", "right": "Don't — binary search is strictly better. Ternary search's real home is finding the peak of a unimodal function, where sorted order doesn't exist." }, { "wrong": "Getting the three-way boundaries wrong.", "right": "The middle third is (m1, m2) exclusive — you already checked both probes. Set lo = m1 + 1 and hi = m2 − 1, or you'll re-check them forever." } ], "quiz": [ { "q": "Ternary search does fewer rounds than binary search. Is it faster?", "options": [ "Yes — fewer rounds means less work", "No — each round costs two comparisons instead of one, so it does more work overall", "They're exactly the same", "Yes, but only on large lists" ], "answer": 1, "why": "Fewer rounds, but double the comparisons per round. It works out to about 26% more comparisons than binary search — slower, despite feeling cleverer." }, { "q": "What is ternary search genuinely good for?", "options": [ "Searching a sorted array", "Finding the peak of a curve that rises then falls", "Sorting", "Nothing at all" ], "answer": 1, "why": "On a unimodal function there is no sorted order to binary-search. Comparing two probe points tells you which side the peak is on — that's a job binary search cannot do." }, { "q": "What's the general lesson here?", "options": [ "Always split into as many parts as possible", "An idea can feel faster and be measurably slower — count, don't guess", "Logarithms are always fast", "Binary search is the only good algorithm" ], "answer": 1, "why": "That's the real takeaway. Intuition proposes; the comparison count decides. Plenty of 'obvious' optimisations are slower once you actually measure them." } ], "problems": [ { "name": "Binary Search", "difficulty": "Easy", "url": "https://leetcode.com/problems/binary-search/" }, { "name": "Peak Index in a Mountain Array", "difficulty": "Medium", "url": "https://leetcode.com/problems/peak-index-in-a-mountain-array/" }, { "name": "Find Peak Element", "difficulty": "Medium", "url": "https://leetcode.com/problems/find-peak-element/" } ], "trace": { "note": "Each frame is one immutable step: data snapshot, pointers, per-index roles, executing code line, and a one-sentence explanation.", "length": 7, "frames": [ { "data": [ { "id": "c0-3", "value": 3 }, { "id": "c1-7", "value": 7 }, { "id": "c2-12", "value": 12 }, { "id": "c3-18", "value": 18 }, { "id": "c4-24", "value": 24 }, { "id": "c5-31", "value": 31 }, { "id": "c6-40", "value": 40 }, { "id": "c7-55", "value": 55 }, { "id": "c8-63", "value": 63 }, { "id": "c9-77", "value": 77 }, { "id": "c10-82", "value": 82 }, { "id": "c11-91", "value": 91 } ], "pointers": [], "highlights": [], "codeLine": 1, "variables": { "target": 82, "size": 12 }, "explanation": "Binary search cuts the window in half. Ternary search cuts it into thirds — surely that's faster? Let's count and find out.", "stats": { "comparisons": 0, "swaps": 0 }, "accent": "active" }, { "data": [ { "id": "c0-3", "value": 3 }, { "id": "c1-7", "value": 7 }, { "id": "c2-12", "value": 12 }, { "id": "c3-18", "value": 18 }, { "id": "c4-24", "value": 24 }, { "id": "c5-31", "value": 31 }, { "id": "c6-40", "value": 40 }, { "id": "c7-55", "value": 55 }, { "id": "c8-63", "value": 63 }, { "id": "c9-77", "value": 77 }, { "id": "c10-82", "value": 82 }, { "id": "c11-91", "value": 91 } ], "pointers": [ { "name": "lo", "index": 0 }, { "name": "m1", "index": 3 }, { "name": "m2", "index": 8 }, { "name": "hi", "index": 11 } ], "highlights": [ { "index": 3, "role": "active" }, { "index": 8, "role": "active" } ], "codeLine": 5, "variables": { "lo": 0, "m1": 3, "m2": 8, "hi": 11, "window": 12 }, "explanation": "Split the window into three with two probes: 18 at 3 and 63 at 8. Note that's *two* looks, not one.", "stats": { "comparisons": 0, "swaps": 0 }, "accent": "compare" }, { "data": [ { "id": "c0-3", "value": 3 }, { "id": "c1-7", "value": 7 }, { "id": "c2-12", "value": 12 }, { "id": "c3-18", "value": 18 }, { "id": "c4-24", "value": 24 }, { "id": "c5-31", "value": 31 }, { "id": "c6-40", "value": 40 }, { "id": "c7-55", "value": 55 }, { "id": "c8-63", "value": 63 }, { "id": "c9-77", "value": 77 }, { "id": "c10-82", "value": 82 }, { "id": "c11-91", "value": 91 } ], "pointers": [], "highlights": [ { "index": 0, "role": "discarded" }, { "index": 1, "role": "discarded" }, { "index": 2, "role": "discarded" }, { "index": 3, "role": "discarded" }, { "index": 4, "role": "discarded" }, { "index": 5, "role": "discarded" }, { "index": 6, "role": "discarded" }, { "index": 7, "role": "discarded" }, { "index": 8, "role": "discarded" } ], "codeLine": 10, "variables": { "lo": 9, "hi": 11 }, "explanation": "82 is bigger than 63, so it can only be in the last third. Drop the first two.", "stats": { "comparisons": 3, "swaps": 0 }, "accent": "discarded" }, { "data": [ { "id": "c0-3", "value": 3 }, { "id": "c1-7", "value": 7 }, { "id": "c2-12", "value": 12 }, { "id": "c3-18", "value": 18 }, { "id": "c4-24", "value": 24 }, { "id": "c5-31", "value": 31 }, { "id": "c6-40", "value": 40 }, { "id": "c7-55", "value": 55 }, { "id": "c8-63", "value": 63 }, { "id": "c9-77", "value": 77 }, { "id": "c10-82", "value": 82 }, { "id": "c11-91", "value": 91 } ], "pointers": [ { "name": "lo", "index": 9 }, { "name": "m1", "index": 9 }, { "name": "m2", "index": 11 }, { "name": "hi", "index": 11 } ], "highlights": [ { "index": 9, "role": "active" }, { "index": 11, "role": "active" }, { "index": 0, "role": "discarded" }, { "index": 1, "role": "discarded" }, { "index": 2, "role": "discarded" }, { "index": 3, "role": "discarded" }, { "index": 4, "role": "discarded" }, { "index": 5, "role": "discarded" }, { "index": 6, "role": "discarded" }, { "index": 7, "role": "discarded" }, { "index": 8, "role": "discarded" } ], "codeLine": 5, "variables": { "lo": 9, "m1": 9, "m2": 11, "hi": 11, "window": 3 }, "explanation": "Split the window into three with two probes: 77 at 9 and 91 at 11. Note that's *two* looks, not one.", "stats": { "comparisons": 3, "swaps": 0 }, "accent": "compare" }, { "data": [ { "id": "c0-3", "value": 3 }, { "id": "c1-7", "value": 7 }, { "id": "c2-12", "value": 12 }, { "id": "c3-18", "value": 18 }, { "id": "c4-24", "value": 24 }, { "id": "c5-31", "value": 31 }, { "id": "c6-40", "value": 40 }, { "id": "c7-55", "value": 55 }, { "id": "c8-63", "value": 63 }, { "id": "c9-77", "value": 77 }, { "id": "c10-82", "value": 82 }, { "id": "c11-91", "value": 91 } ], "pointers": [], "highlights": [ { "index": 0, "role": "discarded" }, { "index": 1, "role": "discarded" }, { "index": 2, "role": "discarded" }, { "index": 3, "role": "discarded" }, { "index": 4, "role": "discarded" }, { "index": 5, "role": "discarded" }, { "index": 6, "role": "discarded" }, { "index": 7, "role": "discarded" }, { "index": 8, "role": "discarded" }, { "index": 9, "role": "discarded" }, { "index": 11, "role": "discarded" } ], "codeLine": 11, "variables": { "lo": 10, "hi": 10 }, "explanation": "82 sits between the two probes, so it must be in the middle third.", "stats": { "comparisons": 6, "swaps": 0 }, "accent": "discarded" }, { "data": [ { "id": "c0-3", "value": 3 }, { "id": "c1-7", "value": 7 }, { "id": "c2-12", "value": 12 }, { "id": "c3-18", "value": 18 }, { "id": "c4-24", "value": 24 }, { "id": "c5-31", "value": 31 }, { "id": "c6-40", "value": 40 }, { "id": "c7-55", "value": 55 }, { "id": "c8-63", "value": 63 }, { "id": "c9-77", "value": 77 }, { "id": "c10-82", "value": 82 }, { "id": "c11-91", "value": 91 } ], "pointers": [ { "name": "lo", "index": 10 }, { "name": "m1", "index": 10 }, { "name": "m2", "index": 10 }, { "name": "hi", "index": 10 } ], "highlights": [ { "index": 10, "role": "active" }, { "index": 10, "role": "active" }, { "index": 0, "role": "discarded" }, { "index": 1, "role": "discarded" }, { "index": 2, "role": "discarded" }, { "index": 3, "role": "discarded" }, { "index": 4, "role": "discarded" }, { "index": 5, "role": "discarded" }, { "index": 6, "role": "discarded" }, { "index": 7, "role": "discarded" }, { "index": 8, "role": "discarded" }, { "index": 9, "role": "discarded" }, { "index": 11, "role": "discarded" } ], "codeLine": 5, "variables": { "lo": 10, "m1": 10, "m2": 10, "hi": 10, "window": 1 }, "explanation": "Split the window into three with two probes: 82 at 10 and 82 at 10. Note that's *two* looks, not one.", "stats": { "comparisons": 6, "swaps": 0 }, "accent": "compare" }, { "data": [ { "id": "c0-3", "value": 3 }, { "id": "c1-7", "value": 7 }, { "id": "c2-12", "value": 12 }, { "id": "c3-18", "value": 18 }, { "id": "c4-24", "value": 24 }, { "id": "c5-31", "value": 31 }, { "id": "c6-40", "value": 40 }, { "id": "c7-55", "value": 55 }, { "id": "c8-63", "value": 63 }, { "id": "c9-77", "value": 77 }, { "id": "c10-82", "value": 82 }, { "id": "c11-91", "value": 91 } ], "pointers": [], "highlights": [ { "index": 10, "role": "found" }, { "index": 0, "role": "discarded" }, { "index": 1, "role": "discarded" }, { "index": 2, "role": "discarded" }, { "index": 3, "role": "discarded" }, { "index": 4, "role": "discarded" }, { "index": 5, "role": "discarded" }, { "index": 6, "role": "discarded" }, { "index": 7, "role": "discarded" }, { "index": 8, "role": "discarded" }, { "index": 9, "role": "discarded" }, { "index": 11, "role": "discarded" } ], "codeLine": 7, "variables": { "result": 10, "looks": 7 }, "explanation": "Found 82 at index 10 after 7 comparisons.", "stats": { "comparisons": 7, "swaps": 0 }, "accent": "found" } ] } }