Data Structures and Algorithms: cram pack

Formula sheets, the key method steps for every topic and 6 practice problems, with full solutions at the back.

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1Formulas and methods

Graph theory

Handshake lemma
Σ deg(v) = 2|E|
Complete graphs
|E(Kn)| = n(n − 1)/2, |E(Km,n)| = m·n
Regular graph
|E| = n·k/2
Tree and forest
|E| = n − 1, |E| = n − c
Leaves of a tree
L = 2 + Σ (deg(v) − 2) over deg(v) ≥ 3
Euler
circuit: all degrees even; path: exactly 2 odd degrees
Hamilton circuits in Kn
(n − 1)!/2
Dijkstra update
d(v) = min(d(v), d(u) + w(u, v))
Walks of length L
(Mᴸ)ij

Method

  1. Write down the degree of every vertex; the degrees add up to twice the number of edges.
  2. For Euler questions, check that the graph is connected and count the vertices with odd degree.
  3. For trees, use n − 1 edges and the handshake lemma.
  4. For Dijkstra, keep a table of tentative distances and settle the smallest unsettled vertex each round.
  5. For a minimum spanning tree, sort the edges by weight and skip every edge that closes a cycle (Kruskal), or grow one tree from a start vertex (Prim).

2Practice problems

1.Algorithms & complexityMedium

A greedy algorithm:

  1. AUses only recursion
  2. BAlways gives the optimal solution
  3. CTries every possible solution
  4. DMakes the locally best choice at each step
2.Algorithms & complexityHard

The halting problem shows that:

  1. AEvery program eventually halts
  2. BSome programs run faster on faster computers, so halting depends on hardware
  3. CLoops should be avoided
  4. DNo algorithm can decide for every program and input whether it halts
3.Data structuresMedium

Which collision resolution method stores colliding keys in a list at the same bucket?

  1. ASeparate chaining
  2. BQuadratic probing
  3. CLinear probing
  4. DDouble hashing
4.Data structuresHard

You must keep the k smallest values from a stream of n numbers using O(k) memory. Which approach works in O(n log k) time?

  1. AA max-heap of size k, replacing its root when a smaller value arrives
  2. BA sorted array of all n values
  3. CA min-heap holding all n values, from which the k smallest are popped at the end
  4. DA FIFO queue of the last k values
5.Graph theoryMedium
ABCDEFEuler circuitEuler path, no circuitneither

Does the connected graph have an Euler circuit, an Euler path that is not a circuit, or neither? Click your answer.

|V| = 6|E| = 6
  1. 1Euler circuit
  2. 2Euler path, no circuit
  3. 3neither
6.Graph theoryHard
21271619139841ABCDEFG

Find the total weight W of a minimum spanning tree of the weighted graph.

|V| = 7|E| = 10
W =

3Solutions

1.Algorithms & complexity

Answer: D · Makes the locally best choice at each step

Why this is the answer

Greedy algorithms are fast and optimal for some problems (e.g. Dijkstra, Huffman coding) but not in general (e.g. coin change with some coin systems).

Answer

D · Makes the locally best choice at each step
2.Algorithms & complexity

Answer: D · No algorithm can decide for every program and input whether it halts

Why this is the answer

Turing showed the halting problem is undecidable: assuming a decider exists leads to a contradiction with a program that does the opposite.

Answer

D · No algorithm can decide for every program and input whether it halts
3.Data structures

Answer: A · Separate chaining

Why this is the answer

Chaining keeps a linked list (or other structure) per bucket; open addressing looks for another free slot in the table itself.

Answer

A · Separate chaining
4.Data structures

Answer: A · A max-heap of size k, replacing its root when a smaller value arrives

Why this is the answer

The max-heap's root is the largest kept value; any smaller new value replaces it in O(log k), so the heap always holds the k smallest so far.

Answer

A · A max-heap of size k, replacing its root when a smaller value arrives
5.Graph theory

Answer: neither

Euler, Hamilton and trees: a connected graph has an Euler circuit (every edge once, back to the start) when every vertex has even degree, and an Euler path that is not a circuit when exactly two vertices have odd degree; the path then runs from one odd vertex to the other. A Hamilton circuit visits every vertex once; Kn has (n − 1)!/2 of them. A tree on n vertices has n − 1 edges, and a connected graph with m edges keeps a spanning tree after removing m − n + 1 edges.

Degrees

deg(A) = 1, deg(B) = 4, deg(C) = 2, deg(D) = 1, deg(E) = 3, deg(F) = 1

Odd vertices

A, D, E, F → 4
neither

Answer

neither
6.Graph theory

Answer: W = 35

Shortest paths and spanning trees: Dijkstra’s algorithm settles one vertex per round: the unsettled vertex with the smallest tentative distance. Its distance is then final, and every neighbour v gets d(v) = min(d(v), d(u) + w(u, v)). Kruskal’s algorithm builds a minimum spanning tree by taking the edges from light to heavy and skipping any edge that would close a cycle; Prim’s algorithm grows one tree from a start vertex by always adding the lightest edge that leaves it. A spanning tree of n vertices has n − 1 edges.

Kruskal’s algorithm

GE (1): take
AB (2): take
GD (4): take
BD (7): take
GC (8): take
GB (9): skip, closes a cycle
AC (12): skip, closes a cycle
EF (13): take

Total weight

W = 1 + 2 + 4 + 7 + 8 + 13 = 35

Answer

W = 35

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