201_optimization_2.pdf - f ac u lty of sci ence d ep ar...

Info icon This preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: f ac u lty of sci ence d ep ar tment of m athemat ics MATH 895-4 Fall 2010 Course Schedule Lecture outline: Optimization algorithms (2) Week Date Sections Part/ References Topic/Sections MACM 201 Discrete Mathematics II Notes/Speaker FS2009 We continuefrom our exploration of optimization algorithms, by looking at algorithms that applies again on 7graphs with weighted and aim at computing a spanning tree of minimum weight. 1 Sept I.1, I.2, I.3 methods Combinatorial edges,Symbolic 2 14 I.4, I.5, I.6 4 28 II.4, II.5, II.6 Structures Unlabelled structures FS: Part A.1, A.2 Applications of such questions are numerous, from computer networks to computational biology, Comtet74 3 21 II.1, II.2, II.3 Labelled structures I Handout #1 through algorithmic theory. (self study) Labelled structures II We will study two algorithms: PrimCombinatorial and Kruskal algorithms. Both are typical examples of Combinatorial 5 Oct 5 III.1, III.2 Asst #1 Due Parameters greedy algorithms: atparameters any given time they make a decision (here to add an edge to an evolving FS A.III 6 12 IV.1, IV.2 Multivariable GFs (self-study) set of edges that, ultimately, will form a spanning tree) that is the best one at the time it is 7 19 IV.3, IV.4 Complex Analysis Analytic Methodsthat it made without trying to ensure is optimal over the course of the whole algorithm. But FS: Part B: IV, V, VI 8 26 Analysis then, we canIV.5prove that thisB4 greedy Singularity decision is actually a good one to make (i.e. it does not Appendix V.1 Stanley 99: Ch. 6 9 us Novaway 2 Due take from finding spanning tree).AsstIn#2 terms of proofs, proving that a greedy Asymptotic methods Handout a #1 minimum (self-study) 9 VI.1 algorithm actually computes an optimal solution to Sophie the problems it aims to solve is not easy, 10 Introduction to Prob. Mariolys but we12 will A.3/ seeC two examples. 18 IX.1 20 IX.2 23 IX.3 25 IX.4 13 30 IX.5 14 Dec 10 11 12 Random Structures and Limit Laws FS: Part C (rotating presentations) Limit Laws and Comb Marni Discrete Limit Laws Sophie Combinatorial instances of discrete Mariolys Continuous Limit Laws Marni Quasi-Powers and Gaussian limit laws Sophie Presentations Asst #3 Due Dr. Marni MISHNA, Department of Mathematics, SIMON FRASER UNIVERSITY Version of: 11-Dec-09 Dr. Cedric Chauve, Spring 2017 Optimization 2: Minimum Spanning Trees 1 f ac u lty of sci ence d ep ar tment of m athemat ics MATH 895-4 Fall 2010 Course Schedule MACM 201 Discrete Mathematics II Week first Date start Sections Part/ References We by defining what isTopic/Sections a minimumNotes/Speaker spanning tree. from FS2009 1 Sept 7 I.1, I.2,weight I.3 Symbolic methods Combinatorial Definition: of a spanning tree. Let G = (V , E) be a graph, w an Structures 2 14 I.4, I.5, I.6 Unlabelled structures Part A.1, A.2 T = (V , E 0 ) be spanning tree of G. edge-weighting ofFS: G, and Comtet74 3 21 II.1, II.2, II.3 Handout #1 (self study) Labelled structures I The weight ofII.6T is the sum of Labelled the weights of the edges in P: 4 28 II.4, II.5, structures II Combinatorial Combinatorial X 5 Oct 5 III.1, III.2 Asst #1 Due parameters Parameters w(P) = w(e) FS A.III 6 12 IV.1, IV.2 (self-study) Multivariable GFs 7 19 IV.3, IV.4 Analytic Methods FS: Part B: IV, V, VI Appendix B4 Stanley 99: Ch. 6 Handout #1 (self-study) Complex Analysis e∈E 0 26 Analysis tree of G whose weight is minimum A8 minimum spanning tree is aSingularity spanning IV.5 V.1 9 Nov 2 Asst #2 Due Asymptotic methods among all spanning trees of G. 10 9 VI.1 Sophie 12 A.3/ C to Prob. Mariolys Problem statement: minimum Introduction spanning tree. 18 IX.1 Limit Laws and Comb Marni 11 Input. A connected graph G = (V , E), V = {v1, . . . , vn}, E = {e1, . . . , em}, Random Structures 20 IX.2 Discrete Limit Laws Sophie and Limit Laws and 23a non-negative edge-weighting of E. Mariolys FS: Part C Combinatorial IX.3 12 (rotating presentations) instances of discrete 25 Continuous Marnithat T = (V , E ) is a minimum Output A IX.4 subset ET of n − 1 edges ofLimitELaws such T Quasi-Powers and 13 30 IX.5 Sophie spanning tree of G. Gaussian limit laws 14 Dec 10 Example. Presentations Asst #3 Due Dr. Marni MISHNA, Department of Mathematics, SIMON FRASER UNIVERSITY Version of: 11-Dec-09 Dr. Cedric Chauve, Spring 2017 Optimization 2: Minimum Spanning Trees 2 MATH 895-4 Fall 2010 Course Schedule f ac u lty of sci ence d ep ar tment of m athemat ics MACM 201 Discrete Mathematics II Week Date Sections Part/ References Notes/Speaker Application 1: approximation forTopic/Sections the Travelling Salesman Problem. from FS2009 The Travelling Problem asks to find a Hamilton path of minimum 1 Sept 7 I.1, I.2, I.3 Salesman Symbolic methods Combinatorial Structures 2 14 I.5, I.6 Unlabelled structures weight. ToI.4,ensure such FS: Part A.1,aA.2Hamiltonian path exists, we assume that the graph G Comtet74 II.1, II.2, II.3 Labelled structures I Handout #1 is 3the21 complete graph Kn. If there are edges we want to exclude for sure from (self study) 4 28 II.4, II.5, II.6 Labelled structures II any Hamilton path, we can just give them a very high weight. 5 Oct 5 III.1, III.2 Example. 6 12 IV.1, IV.2 7 19 8 26 9 Nov 2 IV.3, IV.4 IV.5 V.1 Combinatorial parameters FS A.III (self-study) Combinatorial Parameters Analytic Methods FS: Part B: IV, V, VI Appendix B4 Stanley 99: Ch. 6 Handout #1 (self-study) Complex Analysis Asst #1 Due Multivariable GFs Singularity Analysis Asymptotic methods Asst #2 Due 9 VI.1 12 A.3/ C Introduction to Prob. Mariolys 18 IX.1 Limit Laws and Comb Marni 20 IX.2 Discrete Limit Laws Sophie 23 IX.3 Combinatorial instances of discrete Mariolys 25 IX.4 Continuous Limit Laws Marni 13 30 IX.5 Quasi-Powers and Gaussian limit laws Sophie 14 Dec 10 10 11 12 Random Structures and Limit Laws FS: Part C (rotating presentations) Sophie Presentations Asst #3 Due Dr. Marni MISHNA, Department of Mathematics, SIMON FRASER UNIVERSITY Version of: 11-Dec-09 Dr. Cedric Chauve, Spring 2017 Optimization 2: Minimum Spanning Trees 3 f ac u lty of sci ence d ep ar tment of m athemat ics MATH 895-4 Fall 2010 Course Schedule MACM 201 Discrete Mathematics II Week Date Sections Part/ References Topic/Sections Here again, the problem is difficult: finding aNotes/Speaker Hamiltoni path of minimum weight from FS2009 in1 a Sept non-negatively weighted Symbolic complete graph is NP-complete. There is no 7 I.1, I.2, I.3 methods Combinatorial Structures algorithm that 2 14 I.4, I.5, I.6 will work in a reasonable Unlabelled structurestime for all graphs. FS: Part A.1, A.2 3 21 II.1, II.2, II.3 Comtet74 Handout #1 (self study) Labelled structures I To4 make the problem easier, Labelled we assume that our weighted complete graph 28 II.4, II.5, II.6 structures II satisfies the triangle inequality: for every x,Assty,#1zDue∈ V , w({x, y}) ≤ w({x, z}) + Combinatorial Combinatorial 5 Oct 5 III.1, III.2 parameters Parameters w({y, z}). FS A.III 6 12 IV.1, IV.2 (self-study) Multivariable GFs 7 19 IV.3, IV.4 Now we introduce an Methods algorithmComplex thatAnalysis uses a spanning tree. Analytic 8 26 IV.5 V.1 FS: Part B: IV, V, VI Appendix B4 Stanley 99: Ch. 6 Handout #1 (self-study) Singularity Analysis 1. a Minimum Spanning Tree T of G,Duerooted at an arbitrary vertex; 9 Compute Nov 2 Asst #2 Asymptotic methods 10 9 VI.1 18 IX.1 Sophie 2. Order the vertices of V according to a preorder traversal of T . 12 A.3/ C Introduction to Prob. Mariolys 11 Limit Laws and Comb Marni Claim. LetIX.2 P = xRandom , xn be Discrete the path in GSophie defined by the preorder traversal 20 Limit Laws 1 , . . .Structures and Limit Laws Part C Combinatorial of G 23givenIX.3by theFS: above algorithm. Then w(P) Mariolys is at most twice the cost of a (rotating instances of discrete 12 Hamilton path of presentations) minimum weight in G. In other words, our algorithm is an 25 IX.4 Continuous Limit Laws Marni approximation algorithm for the TSP problem with approximation factor equal Quasi-Powers and 13 30 IX.5 Sophie Gaussian limit laws to142: Dec it 10runs quickly (we will see finding a minimum spanning tree can be done Presentations Asst #3 Due quickly) and the solution it outputs has cost at most twice the cost of a best Hamilton path. Dr. Marni MISHNA, Department of Mathematics, SIMON FRASER UNIVERSITY Version of: 11-Dec-09 Dr. Cedric Chauve, Spring 2017 Optimization 2: Minimum Spanning Trees 4 MATH 895-4 Fall 2010 Course Schedule f ac u lty of sci ence d ep ar tment of m athemat ics MACM 201 Discrete Mathematics II Week Date Sections Example. from FS2009 1 Sept 7 I.1, I.2, I.3 2 14 I.4, I.5, I.6 3 21 II.1, II.2, II.3 4 28 II.4, II.5, II.6 5 Oct 5 III.1, III.2 6 12 IV.1, IV.2 7 19 IV.3, IV.4 8 26 9 Nov 2 IV.5 V.1 Part/ References Topic/Sections Combinatorial Structures FS: Part A.1, A.2 Comtet74 Handout #1 (self study) Symbolic methods Combinatorial parameters FS A.III (self-study) Combinatorial Parameters Analytic Methods FS: Part B: IV, V, VI Appendix B4 Stanley 99: Ch. 6 Handout #1 (self-study) Complex Analysis Notes/Speaker Unlabelled structures Labelled structures I Labelled structures II Asst #1 Due Multivariable GFs Singularity Analysis Asymptotic methods Asst #2 Due 9 VI.1 12 A.3/ C Introduction to Prob. Mariolys 18 IX.1 Limit Laws and Comb Marni 20 IX.2 Discrete Limit Laws Sophie 23 IX.3 Combinatorial instances of discrete Mariolys 25 IX.4 Continuous Limit Laws Marni 13 30 IX.5 Quasi-Powers and Gaussian limit laws Sophie 14 Dec 10 10 11 12 Random Structures and Limit Laws FS: Part C (rotating presentations) Sophie Presentations Asst #3 Due Dr. Marni MISHNA, Department of Mathematics, SIMON FRASER UNIVERSITY Version of: 11-Dec-09 Dr. Cedric Chauve, Spring 2017 Optimization 2: Minimum Spanning Trees 5 MATH 895-4 Fall 2010 Course Schedule f ac u lty of sci ence d ep ar tment of m athemat ics MACM 201 Discrete Mathematics II Week Date Sections Proof of the claim. Part/ References from FS2009 1 Sept 7 I.1, I.2, I.3 2 14 I.4, I.5, I.6 3 21 II.1, II.2, II.3 4 28 II.4, II.5, II.6 5 Oct 5 III.1, III.2 6 12 IV.1, IV.2 7 19 IV.3, IV.4 8 26 9 Nov 2 IV.5 V.1 Topic/Sections Combinatorial Structures FS: Part A.1, A.2 Comtet74 Handout #1 (self study) Symbolic methods Combinatorial parameters FS A.III (self-study) Combinatorial Parameters Analytic Methods FS: Part B: IV, V, VI Appendix B4 Stanley 99: Ch. 6 Handout #1 (self-study) Complex Analysis Notes/Speaker Unlabelled structures Labelled structures I Labelled structures II Asst #1 Due Multivariable GFs Singularity Analysis Asymptotic methods Asst #2 Due 9 VI.1 12 A.3/ C Introduction to Prob. Mariolys 18 IX.1 Limit Laws and Comb Marni 20 IX.2 Discrete Limit Laws Sophie 23 IX.3 Combinatorial instances of discrete Mariolys 25 IX.4 Continuous Limit Laws Marni 13 30 IX.5 Quasi-Powers and Gaussian limit laws Sophie 14 Dec 10 10 11 12 Random Structures and Limit Laws FS: Part C (rotating presentations) Sophie Presentations Asst #3 Due Dr. Marni MISHNA, Department of Mathematics, SIMON FRASER UNIVERSITY Version of: 11-Dec-09 Dr. Cedric Chauve, Spring 2017 Optimization 2: Minimum Spanning Trees 6 f ac u lty of sci ence d ep ar tment of m athemat ics MATH 895-4 Fall 2010 Course Schedule MACM 201 Discrete Mathematics II Week Date Sections Part/ References Topic/Sectionsoutbreak. Notes/Speaker Application 2: analysis of a pathogen from FS2009 In1 a work I am doing with BC Center for Disease Control, we have Sept 7 I.1, I.2, I.3 currently Symbolic the methods Combinatorial Structures 2 14 I.4,the I.5, I.6 genomes of the bacteria Unlabelled structures sequenced responsible for the disease tuberculosis FS: Part A.1, A.2 Comtet74 3 21 II.1, II.2, II.3 Labelled structures I Handout #1 (Mycobacterium tuberculosis) taken from a few hundred of patients of an out(self study) 4 28 II.4, II.5, II.6 Labelled structures II break of tuberculosis (TB) in BC. We want to see if we can cluster the patients Combinatorial Combinatorial 5 Oct 5 III.1, III.2 Asst #1 Due Parameters into groups, whereparameters in each group, all patients are likely to have been infected FS A.III 6 12 IV.1, IV.2 Multivariable GFs (self-study) with the same strain of the pathogen (i.e. the same source). 7 19 IV.3, IV.4 Analytic Methods FS: Part B: IV, V, VI Appendix B4 Stanley 99: Ch. 6 Handout #1 (self-study) Complex Analysis This has important applicationsSingularity as not all strains of TB are treated the same 8 26 Analysis IV.5 V.1 9 Nov 2 Asst #2 Due and need to be treated with way: some are resistant to first-line antibiotics Asymptotic methods 9 VI.1 Sophie stronger antiobtics, but then you want to be careful that they do not acquire 10 12 A.3/ C Introduction to Prob. Mariolys resistance to these stronger antibiotics and then infect other patients other 18 IX.1 Limit Laws and Comb Marni 11 resistance spreads in the population and this is a very serious problem. Random Structures 20 IX.2 Discrete Limit Laws Sophie and Limit Laws FS: Part C (rotating presentations) Combinatorial 23 IX.3 Mariolys The techniques we use rely on advanced technology (to sequence the genomes of instances of discrete 12 25 IX.4 Continuous Limit Laws Marni the pathogen) and advanced tools in computational biology (to detect mutations Quasi-Powers and in13some – IX.5afew hundred – selected genes ofSophie Mycobacterium tuberculosis in the 30 Gaussian limit laws patients). But once wePresentations have this set of mutations 14 Dec 10 Asst #3 Due for all our patients, we use two simple tools we did see in MACM201: the Hamming distance and minimum spanning trees. Dr. Marni MISHNA, Department of Mathematics, SIMON FRASER UNIVERSITY Version of: 11-Dec-09 Dr. Cedric Chauve, Spring 2017 Optimization 2: Minimum Spanning Trees 7 MATH 895-4 Fall 2010 Course Schedule f ac u lty of sci ence d ep ar tment of m athemat ics MACM 201 Discrete Mathematics II Week do Date Sections Part/ References We thefrom following: FS2009 1 Sept 7 I.1, I.2, I.3 Topic/Sections Notes/Speaker Symbolic methods Combinatorial • We build a complete graph G = (V , E) where each vertex E represents Structures 2 14 I.4, I.5, I.6 Unlabelled structures FS: Part A.1, A.2 a patient and the edge between two vertices x and y is weighted by the Comtet74 3 21 II.1, II.2, II.3 Labelled structures I Handout #1 (self study) between distance the genomes of Mycobacterium tuberculosis 4 Hamming 28 II.4, II.5, II.6 Labelled structures II sequenced in Combinatorial both patients.Combinatorial 5 Oct 5 III.1, III.2 Asst #1 Due parameters FS A.III (self-study) 6 Parameters 12 IV.1, IV.2 Multivariable GFs • We compute a minimum spanning tree in this graph, and from this tree, we 7 19 IV.3, IV.4 Complex Analysis Methods can easily seeAnalytic (visualize) potential clusters of patients infected by pathogens FS: Part B: IV, V, VI 8 26 Singularity Analysis Appendix B4 IV.5genomes V.1 whose Stanleyhave 99: Ch. 6 very similar genomes, forming the clusters we are 9 Nov 2 Asst #2 Due Asymptotic methods Handout #1 looking for. (self-study) 9 VI.1 Sophie 10 12 A.3/ C Introduction to Prob. Mariolys 18 IX.1 Limit Laws and Comb Marni 20 IX.2 Discrete Limit Laws Sophie 23 IX.3 Combinatorial instances of discrete Mariolys 25 IX.4 Continuous Limit Laws Marni 13 30 IX.5 Quasi-Powers and Gaussian limit laws Sophie 14 Dec 10 11 12 Random Structures and Limit Laws FS: Part C (rotating presentations) Presentations Asst #3 Due Dr. Marni MISHNA, Department of Mathematics, SIMON FRASER UNIVERSITY Version of: 11-Dec-09 Dr. Cedric Chauve, Spring 2017 Optimization 2: Minimum Spanning Trees 8 MATH 895-4 Fall 2010 Course Schedule f ac u lty of sci ence d ep ar tment of m athemat ics MACM 201 Discrete Mathematics II Week Date Sections Part/ References Topic/Sections Notes/Speaker Algorithmic principles. from FS2009 We to look at two algorithms: 1 are Sept 7going I.1, I.2, I.3 Symbolic methodsPrim’s algorithm and Kruskal algorithm. Combinatorial Structures 2 14 I.4, I.5, I.6 Unlabelled structures FS: Part A.1, A.2 Both algorithms are greedy algorithm: Comtet74 3 21 II.1, II.2, II.3 Handout #1 (self study) Labelled structures I 4 they 28 II.4, II.5, II.6 structures IIE 1. want to compute a setLabelled of edges T that forms a spanning tree of G, 5 Oct 5 III.1, III.2 6 12 IV.1, IV.2 Combinatorial parameters FS A.IIIT (self-study) 2. they start with E empty, Combinatorial Parameters Asst #1 Due Multivariable GFs 7 they 19 IV.3, IV.4 Complex 3. repeatedly (n−1 if GAnalysis has n vertices) add one edge to the current Analytic Methodstimes FS: Part B: IV, V, VI 8 26 Singularity Analysis Appendix B4 set ETIV.5of edges, V.1 9 Stanley 99: Ch. 6 Handout #1 (self-study) Nov 2 Asymptotic methods Asst #2 Due 4. Kruskal’s algorithm adds an edge that does 9 VI.1 Sophienot create a cycle, of min. weight 12 A.3/all C Introduction to Prob. Mariolys among such possible edges, 10 11 18 IX.1 Limit Laws and Comb Marni 5. Prim’s adds an edge connected Random Structures 20 IX.2algorithm Discrete Limit Laws Sophieto ET , of min. weight among all and Limit Laws FS: Part C Combinatorial such possible edges. 23 IX.3 Mariolys 12 25 IX.4 (rotating presentations) instances of discrete Continuous Limit Laws Marni Quasi-Powers andof G with n vertices, n − 1 edges and Kruskal’s algorithm builds a subgraph 13 30 IX.5 Sophie Gaussian limit laws that is acyclic: a spanning tree. 14 Dec 10 Presentations Asst #3 Due Prim’s algorithm builds a subgraph of G with n vertices, n − 1 edges and that is connected: a spanning tree. The key question is: why this approach to take at any time the best available edge ensures the resulting spanning tree is of minimum weight among all spanning trees? Dr. Marni MISHNA, Department of Mathematics, SIMON FRASER UNIVERSITY Version of: 11-Dec-09 Dr. Cedric Chauve, Spring 2017 Optimization 2: Minimum Spanning Trees 9 MATH 895-4 Fall 2010 Course Schedule f ac u lty of sci ence d ep ar tment of m athemat ics MACM 201 Discrete Mathematics II Week Date Sections Part/ References Topic/Sections Notes/Speaker Kruskal’s algorithm. from FS2009 Input: = (V , E) withSymbolic |V |methods = n. 1 Septa 7 graph I.1, I.2, I.3 G Combinatorial 2 14 I.5, I.6 Unlabelled structures Output: aI.4,subet EStructures ofA.1,EA.2that forms a minimum weight spanning tree of G. FS: T Part 3 21 II.1, II.2, II.3 1. Initialization. 28 II.4, II.5, II.6 4 5 6 7 11 12 Combinatorial Parameters 12 IV.1, IV.2 (self-study) Multivariable GFs 19 IV.3, IV.4 Analytic Methods FS: Part B: IV, V, VI Appendix B4 Stanley 99: Ch. 6 Handout #1 (self-study) Complex Analysis Nov 2 Asst #1 Due Singularity Analysis Asymptotic methods Asst #2 Due Quasi-Powers and Gaussian limit laws Sophie (a) For 9 VI.1 i from 1 to n − 2 Do /* Assume Sophie ET = {e1 , . . . , ei } */ 12 C i.A.3/Let e be an edge ofIntroduction E − EtoTProb.thatMariolys satisfies 18 IX.1 Limit Laws and Comb Marni A. (V , E {e}) is Discrete acyclic, T ∪Structures Random 20 IX.2 Limit Laws Sophie and Limit Laws C B. e isFS:ofPartmin.weight among all edges of E − ET that satisfies A. Combinatorial 23 IX.3 Mariolys (rotating instances of discrete ii.IX.4Add e presentations) to ET and denote by . 25 Continuousit Limit Lawsei+1 Marni 13 30 IX.5 Illustration. 14 Labelled structures II (b) Add to ET an edge of E of minimum weight. Call it e1. 26 2. Main loop. IV.5 V.1 10 Labelled structures I Combinatorial Oct 5 III.1, III.2 (a) Set ET =parameters ∅. FS A.III 8 9 Comtet74 Handout #1 (self study) Dec 10 Presentations Asst #3 Due Dr. Marni MISHNA, Department of Mathematics, SIMON FRASER UNIVERSITY Version of: 11-Dec-09 Dr. Cedric Chauve, Spring 2017 Optimization 2: Minimum Spanning Trees 10 MATH 895-4 Fall 2010 Course Schedule f ac u lty of sci ence d ep ar tment of m athemat ics MACM 201 Discrete Mathematics II Week Date Sections Part/ References Topic/Sections Notes/Speaker Prim’s algorithm. from FS2009 Input: = (V , E) withSymbolic |V |methods = n. 1 Septa 7 graph I.1, I.2, I.3 G Combinatorial 2 14 I.5, I.6 Unlabelled structures Output: aI.4,subet EStructures ofA.1,EA.2that forms a minimum weight spanning tree of G. FS...
View Full Document

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern