{ "openapi": "3.1.1", "info": { "title": "Big-O Crossover Calculator API", "version": "1.0.0", "summary": "When does c₁·n log n beat c₂·n²? Crossover points, log-log chart and run times.", "description": "Every endpoint performs the same calculation as the human page and returns inputs, result, units, assumptions, sources, warnings and computedAt. MCP endpoint: /mcp." }, "servers": [ { "url": "https://bigocrossovercalculator.usefulweb.site" } ], "paths": { "/api/v1/calculate": { "post": { "operationId": "calculate", "summary": "When does c₁·n log n beat c₂·n²? Crossover points, log-log chart and run times.", "description": "Compares cost functions such as 50·n log n against 2·n² or 2ⁿ with your constant factors, finds every input size where one overtakes another, draws them on a log-log chart, and converts operation counts into running time at a given number of operations per second.\n\nMethodology:\n- Each cost is c·f(n) with log₂ for logarithms; n! is evaluated as Γ(n + 1), so every function is continuous in n. All comparisons are done on log10 of the cost, so 2ⁿ or n! for large n never overflow.\n- Crossovers: the difference of the log costs is sampled at 2,000 log-spaced sizes between the smallest and largest n; every sign change is refined by 80 bisection steps. Crossovers closer together than the grid spacing can be missed.\n- Time = cost ÷ operations per second. Real programs also depend on memory access, caches and constant overheads that a single constant c only approximates.", "tags": [ "math" ], "x-mcp-tool": "big_o_crossover_calculator", "requestBody": { "required": true, "content": { "application/json": { "schema": { "$ref": "#/components/schemas/BigOCrossoverCalculatorInput" }, "examples": { "example1": { "summary": "Merge sort vs insertion sort", "value": { "functions": [ { "label": "Merge sort", "complexity": "nlogn", "constant": 6 }, { "label": "Insertion sort", "complexity": "n2", "constant": 0.5 } ] } }, "example2": { "summary": "Brute force 2ⁿ vs n³ dynamic programming", "value": { "functions": [ { "label": "Brute force", "complexity": "2n", "constant": 1 }, { "label": "DP", "complexity": "n3", "constant": 100 }, { "label": "Fancy n^2.37", "complexity": "nk", "constant": 1000000, "exponent": 2.373 } ], "nMax": 100000, "sizes": [ "10", "20", "30", "50", "100", "1000" ] } } } } } }, "responses": { "200": { "description": "Calculation result envelope.", "content": { "application/json": { "schema": { "$ref": "#/components/schemas/BigOCrossoverCalculatorResult" } } } }, "400": { "description": "Malformed JSON body.", "content": { "application/problem+json": { "schema": { "$ref": "#/components/schemas/Problem" } } } }, "422": { "description": "Invalid input (see errors[]).", "content": { "application/problem+json": { "schema": { "$ref": "#/components/schemas/Problem" } } } }, "429": { "description": "Rate limit exceeded (see Retry-After).", "content": { "application/problem+json": { "schema": { "$ref": "#/components/schemas/Problem" } } } } } } }, "/api/v1/tool": { "get": { "operationId": "getToolDefinition", "summary": "Inputs, output schema and examples for this tool", "responses": { "200": { "description": "Tool definition.", "content": { "application/json": { "schema": { "type": "object" } } } } } } } }, "components": { "schemas": { "Problem": { "type": "object", "description": "RFC 9457 problem details. 422 responses include per-field `errors`.", "required": [ "type", "title", "status", "detail" ], "properties": { "type": { "type": "string" }, "title": { "type": "string" }, "status": { "type": "integer" }, "detail": { "type": "string" }, "code": { "type": "string" }, "errors": { "type": "array", "items": { "type": "object", "required": [ "field", "message" ], "properties": { "field": { "type": "string" }, "message": { "type": "string" } } } } } }, "BigOCrossoverCalculatorInput": { "type": "object", "additionalProperties": false, "properties": { "functions": { "title": "Cost functions", "type": "array", "minItems": 1, "maxItems": 8, "items": { "type": "object", "additionalProperties": false, "properties": { "label": { "title": "Name", "type": "string", "maxLength": 40 }, "complexity": { "title": "Growth", "type": "string", "enum": [ "1", "logn", "sqrtn", "n", "nlogn", "n2", "n3", "nk", "2n", "nfact" ], "default": "n" }, "constant": { "title": "Constant c", "type": "number", "minimum": 1e-12, "maximum": 1000000000000000, "default": 1 }, "exponent": { "title": "k (for n^k)", "type": "number", "minimum": 0, "maximum": 20 } }, "required": [ "complexity", "constant" ] }, "default": [ { "label": "Merge sort", "complexity": "nlogn", "constant": 6 }, { "label": "Insertion sort", "complexity": "n2", "constant": 0.5 } ] }, "nMin": { "title": "Smallest n", "type": "number", "minimum": 1, "maximum": 1000000000000, "default": 2 }, "nMax": { "title": "Largest n", "type": "number", "minimum": 2, "maximum": 1000000000000000, "default": 1000000 }, "opsPerSecond": { "title": "Operations per second", "description": "For converting cost into time; 10^9 is a rough figure for simple operations on one CPU core.", "type": "number", "minimum": 1, "maximum": 1000000000000000000, "default": 1000000000 }, "sizes": { "title": "Input sizes for the table", "type": "array", "items": { "type": "string" }, "maxItems": 12, "default": [ "10", "100", "1000", "10000", "100000", "1000000" ] } }, "required": [ "functions" ] }, "BigOCrossoverCalculatorResult": { "type": "object", "additionalProperties": false, "required": [ "schemaVersion", "tool", "computedAt", "inputs", "summary", "result", "units", "assumptions", "sources", "warnings", "presentation", "links" ], "properties": { "schemaVersion": { "type": "string", "const": "1.0" }, "tool": { "type": "object", "additionalProperties": false, "required": [ "id", "name", "version" ], "properties": { "id": { "type": "string" }, "name": { "type": "string" }, "version": { "type": "string" } } }, "computedAt": { "type": "string", "format": "date-time" }, "inputs": { "type": "object", "description": "Normalised inputs actually used, including applied defaults." }, "summary": { "type": "string" }, "result": { "type": "object", "required": [ "functions", "crossovers", "table", "opsPerSecond" ], "properties": { "opsPerSecond": { "type": "number" }, "functions": { "type": "array", "items": { "type": "object", "required": [ "label", "formula" ], "properties": { "label": { "type": "string" }, "formula": { "type": "string" } } } }, "crossovers": { "type": "array", "items": { "type": "object", "required": [ "between", "n", "cheaperAbove" ], "properties": { "between": { "type": "array", "items": { "type": "string" } }, "n": { "type": "number" }, "firstIntegerN": { "type": "integer" }, "cheaperAbove": { "type": "string" } } } }, "table": { "type": "array", "items": { "type": "object", "required": [ "n", "costs" ], "properties": { "n": { "type": "number" }, "costs": { "type": "array", "items": { "type": "object", "required": [ "label", "log10Ops", "ops", "time" ], "properties": { "label": { "type": "string" }, "log10Ops": { "type": [ "number", "null" ] }, "ops": { "type": "string" }, "time": { "type": "string" } } } } } } }, "cheapestAtMax": { "type": "string" } } }, "units": { "type": "object", "additionalProperties": { "type": "string" } }, "assumptions": { "type": "array", "items": { "type": "string" } }, "sources": { "type": "array", "items": { "type": "object", "additionalProperties": false, "required": [ "title" ], "properties": { "title": { "type": "string" }, "url": { "type": "string" }, "publisher": { "type": "string" }, "retrievedAt": { "type": "string" }, "note": { "type": "string" } } } }, "warnings": { "type": "array", "items": { "type": "string" } }, "presentation": { "type": "array", "items": { "type": "object" }, "description": "Display blocks used by the human page (same data)." }, "links": { "type": "object", "additionalProperties": false, "required": [ "human", "api", "openapi", "methodology" ], "properties": { "human": { "type": "string", "format": "uri" }, "api": { "type": "string", "format": "uri" }, "openapi": { "type": "string", "format": "uri" }, "methodology": { "type": "string", "format": "uri" } } } } } } } }