Posts match “ algorithm ” tag:

[Styletrip] 在總旅程時間限制下，如何有效的找到可行的旅遊路線

This is a paper study note for my startup travel service called StyleTrip.
The paper is Trip-Mine: An Efficient Trip Planning Approach with Travel Time Constraints

Introduction

• How to efficiently search for the most interesting trip which satisfies a travel time constraint from a large number of attractions.
• The efficient of trip planning is sensitive to the scalability of travel regions.
• The optimal trip finding problem is different from Traveling Sales Problem(TSP).
• The reason is that TSP finds the shortest path from the extra N given locations. However, the optimal trip problem is to finds a sub-permutation of given N locations.
• The dynamic programming searches all possible trips based on the concept of DFS.

Optimization mechanisms

• Attraction sorting ( User favorite, the must-go / popular / high-rating places )
• Low bound checking ( meeting time, cost constaints )
• Score Checking ( meeting low bound checking and the score is maximum）

Definitions

• Score: It can refer to user favorite , popularity or rating value.
• Valid trip: A trip is valid if the total trip time is less than or equal to the required travel time.
• Optima trip: A valid trip is optimal if the Score(tripA) is maximum among the valid trip.

Proposed theorem

• We can find out that calculation of the trip takes an exponential time, so that it can conclude that this is a NP complete problem.
• Triangle inequality Travel time (x,y)+(y,x) >= (x,z)
• If travel time (a0,trip1) <= (a0,trip2), then trip1 is the sub-trip of trip2.
• Given a invalid (total trip time exploded) trip1, then any super-trip of trip1 is invalid, too. (pruning strategy 1)

Pruning strategy

1. Given a invalid (total trip time exploded) trip1, then any super-trip of trip1 is invalid, too. Based on the pruning strategy 1, we can prune the trips which contain any invalid trip to reduce the calculations.
2. Given an place set A, A is not a valid trip set if the trip time is greater than the travel time constraint.

Approach

1. Find the candidate place set: starting from 1-place-set, 2-place-set (#2 permutation) ... , n-place-set (#n combination), we find all valid trip permutation and remove all the invalid trip and its super-trip permuation (pruning strategy 1).
2. Filter the valie trip set with pruning strategy 2.
3. The trip scores of a trip set and its all permutations are the same, so we don't need to check the permutations which satisifies the travel time constraint.

[Discrete Optimization Note] 2. Knapsack problem

This is my note for Discrete Optimization in Coursera (https://class.coursera.org/optimization-002)

Greedy Algorithm Intuition

What is Greedy? Build a solution by picking itmes one at a time in greedy fasion.

Idea 3: value density!

Calculate dollars per kilogram.

Greedy Algorithm

• For one problem, there are many possible greddy algorithms. Some will do better than others. (Take a look at above ideas)
• Idea: Take the most valuable items first. And the next valuable item that can fit into a knapsack.
1. Quick to design and implement the algorithm.
2. It can be very fast to model the problem.
• Problems:
1. No quality guarantees (in general). You don't know much you can improve the algorithm, you don't know how good they are.
2. Quality can vary widely on the input.

Summary of Greedy Algorithm:

Try to understand what the problem is. Get a base line, that's what you have to improve and then afterwards you going to use some of the techniques that we will present:

• Constraint Programming
• Local Searching
• Mixed Integer Programming To improve the greedy algorithm and find out how much you can improve.

Modeling

How to formalize an optimization task as a mathematical model. This is the key to solve the optimal problem.

How to model an optimization problem

1. Describe the problem
2. Choose some decision variables
3. Express the problem constraints in term of there variables
4. Express the objective function ( the quality of the solution )
5. There may be many ways to model an optimization problem.

Modeling the knapsack problem

1. Describe the problem:
• For each i in Item, has weight Wi and value Vi.
• We have a knapsack which capacity is K.
• We have to find the subset of Item which value is maximum and summation of weight is less than or equal to K.
2. Choose some decision variables:
• Xi is item i is selected or not
3. Express the problem constraints in term of there variables:
4. Express the objective function ( the quality of the solution ) is maximum and be subject to
5. There may be many ways to model an optimization problem.

Dynamic Programming

Basic Principle:

• divide and conquer
• bottom up computation

Define the problem:

• Item selection I = { 1, 2, 3 ...... n }
• O( k, j ) is the optimal solution of capacity K of Item[ 1 ... j ] s.t.

maximize

and subject to

Recurrence relation

• Before we want to solve O( k, j ), we solve sub-problemO( k, j - 1 ) first.

一、 Describe the problem

系統目標

• 輸入: 使用者輸入地點關鍵字(想去的地方)，和去的日期(何時去，選填)
• 輸出: 引擎計算出搜尋結果的地點子集合，且該子集合為有順序性，同時滿足下列條件:
1. 包含想去的區域內最熱門、必去的景點集合
2. 子集合內的景點都是使用者最有興趣的
3. 所有要去的地點花費總時間滿足時間限制
4. 所有要去的地點在要去的期間內都有開放或營業
5. 三餐有建議的餐飲，晚上有住宿
6. 交通總時間要最小

Symbolization

• For each place p in {PLACE}, has search rank R,place category pc, place weight pw , popularity & rating pr, open day 'od', open time ot, routing time rt, and spend time st.
• For the user u, has place keywords keywords, travel dates dates (total days=D) and user favorite uf.
• We have to find the subset trip of {PLACE}, which search rank, popularity, user favorite is maximum, the summation of routing time + spend time is less than and equal to D.

二、 Choose some decision variables

Xi 表示地點p 有選擇或沒有選擇到排程當中。

三、 Express the problem constraints in term of there variables

1. 所有要去的地點花費總時間滿足時間限制
2. 所有要去的地點在要去的期間內都有開放或營業
3. 三餐有建議的餐飲，晚上有住宿

四、 Express the objective function ( the quality of the solution )

1. 包含想去的區域內最熱門、必去的景點集合
2. 子集合內的景點都是使用者最有興趣的
3. 整個交通時間要最小

前情提要

1. 基於機率型分類（probabilistic classification
2. 使用貝氏定理來做計算。
3. 假設特徵之間事件獨立（independence)。
4. 最常用於文件自動分類的應用上。

演算法思路

貝氏定理

• prior probability: 先驗機率，P(A)是在觀察特徵或證據「前」，不考慮任何事件B的因素，能表達事件A的機率分佈。
• posterior probability: 後驗機率，P(A|B)是在觀察特徵或證據「後」，所得到的條件機率。

訓練階段

C63發表會 P 15 25 0 5 8 3
BMW i8 P 35 40 1 3 3 2

Android 5.0 T 10 5 7 0 2 30
iPhone6 T 5 5 5 15 8 32

字詞分佈模式

• Bernouli: 只判斷字詞是否有出現，就出現就是1，沒有出現就是0。
• P(分類) = 該分類新聞篇數 / 所有訓練集合新聞篇數
• P(特徵關鍵字|分類) = (該分類下包含特徵關鍵字的新聞篇數 + 1) / (該分類下包含特徵關鍵字的新聞篇數 + 2)
• Multinomial: 直接採用字詞出現頻率。
• P(分類) = 該分類下字詞頻率總和 / 所有訓練集合字詞頻率總和
• P(特徵關鍵字|分類) = (該分類下、該關鍵字字詞頻率總和 + 1) / (該分類下所有關鍵字字詞頻率總和 + 訓練集合關鍵字個數)

結論

• 貝氏分類對於少量的訓練集合一樣會有不錯的分類準度，它的威力恰好在於小樣本下，專家意見或歷史經驗，也就是所謂的先驗分配，能夠補足小樣本的不足，使得推論能夠進行。
• 適合用在資料會不斷成長的應用。