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. ATries every possible solution
  2. BMakes the locally best choice at each step
  3. CAlways gives the optimal solution
  4. DUses only recursion
2.Algorithms & complexityHard

Shortest simple paths can be found in polynomial time. What is known about finding the longest simple path between two vertices in a general graph?

  1. AIt can be solved by Dijkstra's algorithm with negated weights
  2. BIt is undecidable
  3. CIt is NP-hard
  4. DIt is always solvable in O(V + E)
3.Data structuresMedium

In a binary min-heap stored in an array (index starting at 0), the children of the node at index i are at:

  1. A2i + 1 and 2i + 2
  2. B2i and 2i + 1
  3. Ci/2 and i/2 + 1
  4. Di + 1 and i + 2
4.Data structuresHard

In a hash table using open addressing, why is deletion often done with a "tombstone" marker?

  1. AEmptying the slot could break the probe sequence for other keys
  2. BIt makes the table sorted
  3. CDeleted keys must be kept forever for security, so slots can never be reused
  4. DTombstones speed up hashing
5.Graph theoryMedium
ABCDEEuler 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| = 5|E| = 5
  1. 1Euler circuit
  2. 2Euler path, no circuit
  3. 3neither
6.Graph theoryHard
191671112563ABCDEF

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

|V| = 6|E| = 9
W =

3Solutions

1.Algorithms & complexity

Answer: B · 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

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

Answer: C · It is NP-hard

Why this is the answer

A longest simple path of length V − 1 is a Hamiltonian path, an NP-complete problem; on directed acyclic graphs, however, it is solvable in linear time.

Answer

C · It is NP-hard
3.Data structures

Answer: A · 2i + 1 and 2i + 2

Why this is the answer

For 0-based arrays, the left child is at 2i + 1, the right child at 2i + 2, and the parent at ⌊(i − 1)/2⌋.

Answer

A · 2i + 1 and 2i + 2
4.Data structures

Answer: A · Emptying the slot could break the probe sequence for other keys

Why this is the answer

If a deleted slot became empty, later searches might stop early and miss keys that were placed further along the probe chain.

Answer

A · Emptying the slot could break the probe sequence for other keys
5.Graph theory

Answer: Euler circuit

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) = 2, deg(B) = 2, deg(C) = 2, deg(D) = 2, deg(E) = 2

Odd vertices

0
Euler circuit

Answer

Euler circuit
6.Graph theory

Answer: W = 24

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

AB (1): take
BE (3): take
DF (5): take
EF (6): take
BD (7): skip, closes a cycle
AC (9): take

Total weight

W = 1 + 3 + 5 + 6 + 9 = 24

Answer

W = 24

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