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<slide-template title="Optimizing Transaction Schedules on Universal Quantum Computers via Code Generation for Grover's Search Algorithm" subtitle=""></slide-template>

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<title-slide titletype="IDEAS 2021" title="" modulenumber="July 14-16, 2020, Montreal, QC, Canada" subtitle="Optimizing Transaction Schedules on Universal Quantum Computers via Code Generation for Grover's Search Algorithm"></title-slide>

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# Using .darkblue[Hardware Accelerator] for optimizing Transaction Schedules
.reference[T. Bittner, S. Groppe, Avoiding Blocking by Scheduling Transactions using Quantum Annealing, IDEAS 2020]

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# .small-font32.bold[.darkblue[2 Phase Locking (2PL)] versus .darkblue[Strict Conservative 2PL]]
.reference[T. Bittner, S. Groppe, Avoiding Blocking by Scheduling Transactions using Quantum Annealing, IDEAS 2020]

.more-condensed[
- .darkblue[required locks] to be determined by
  - .darkblue[static analysis] of transaction, .darkgray[or if static analysis is not possible:]
  - an .darkblue[additional phase at runtime] before transaction processing
      - <span class="darkgray">A. Thomson et al., "Calvin: Fast distributed transactions for partitioned database systems", SIGMOD 2012.</span>
]

---

# .em[Optimizing] .darkblue[Transaction Schedules]

.reference[$^*$ M.R. Garey, D.S. Johnson and R. Sethi, The Complexity of Flowshop and Jobshop Scheduling, 1(2):117-129, 1976]

.more-more-condensed[
- Variant of job shop schedule problem (JSSP):
  - Multi-Core CPU
      - Process whole <span class="darkblue">job (here transaction) on core X</span>
  - .darkblue[Schedule: $\forall$ cores: Sequence of jobs] to be processed
  - What is the .darkblue[optimal schedule] for minimal overall processing time? 
- .darkgray[Additionally to JSSP:]<br/>.darkblue[Blocking transactions not] to be processed .darkblue[in parallel]
- Example:
<table class="fullwidth">
<tr>
<td style="width:30%"><img src="../../img/keynotes/BlockingTransactions.svg" align="top" width="100%"></img><div class="small-font18 textcenter">Black: Blocking transactions</div></td>
<td style="width:5%"></td>
<td style="width:30%"><img src="../../img/keynotes/DistributionOfTransactions.svg" align="top" width="100%"></img><div class="small-font18 textcenter">Transaction schedule</div></td>
<td style="width:35%">.more-more-condensed[
- JSSP is among the .darkblue[hardest combinatorial optimizing problems]$^*$
- $\Rightarrow$ .darkblue[Hardware accelerating] the optimization of transaction schedules
]</td>
</tr>
</table>
]

---

# Architectures of .darkblue[Emergent Hardware]
.reference[Extended from <a target="_blank" href="http://www.imprs-dynamics.mpg.de/pdfs/Plessl_talk.pdf">C. Plessl, Accelerating Scientific Computing with Massively Parallel Computer Architectures, IMPRS Winter School, Wroclaw, 2012</a>]

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# <img src="../../img/keynotes/QuantumComputer.svg" align="top" width="10%"></img> .darkblue[Quantum Computer]

.condensed[
<ul>
<li> use of .darkblue[quantum-mechanical phenomena such as superposition and entanglement] to perform computation</li>
<li> Different types of quantum computer, e.g.
  <ul><li> .darkblue[Universal Quantum Computer]
      <ul><li> uses <span class="darkblue">quantum logic gates</span> arranged in a circuit to do computation</li>
	      <li> <span class="darkblue">measurement</span> (sometimes called observation) assigns the observed variable to a single value</li>
      </ul></li>
  <li> .darkblue[Quantum Annealing]
      <ul><li> <span class="darkblue">metaheuristic for finding the global minimum</span> of a given objective function over a given set of candidate solutions</li>
          <li> i.e., some way to solve a special type of <span class="darkblue">mathematical optimization problem</span></li>
      </ul>
  </li>
</ul>
]

---

# .darkblue[Grover]'s Search Algorithm<br/>

.more-more-condensed[
- .darkblue[Black box] function $f:${$0,...,2^b -1$} $\mapsto$ {$true, false$}
- Grover's search algorithm finds one $x \in${$0,...,2^b -1$},<br/>such that $f(x)=true$
  - if there is only .darkblue[one solution]: $\frac{\pi}{4}\cdot\sqrt{2^b}$ basic steps each of which calls $f$<br/>Let $f'(b)$ be runtime complexity of $f$ for testing $x$ to be true:<br/>.darkblue[$\Rightarrow O(\sqrt{2^b}\cdot f'(b))$]
  - if there are .darkblue[$k$] possible .darkblue[solutions]: .darkblue[$O(\sqrt{\frac{2^b}{k}}\cdot f'(b))$]
]

---

# .darkblue[Overview] of Optimizing Transaction Schedules via Quantum Computing

---

# .darkblue[Encoding Scheme] .small-font36[of Transaction Schedules]

<div class="textcenter">
<img src="../../img/ideas21/AlgoDetermineSchedule.jpg" align="top" width="70%"></img>
</div>
.more-more-condensed[
- $29 =$ .darkgreen[$000111$] .darkred[$01$] $_{binary} \equiv$ Core 0 .darkgreen[$[3,1]$] .darkred[$\mu_1=1$] .darkgreen[$[0,2]$] Core 1, some bits for .darkgreen[permutation] and some for .darkred[separators] 
- $b = (m-1)\cdot \left \lceil log_2(n-1)\right \rceil + \left \lceil log_2(n!-1)\right \rceil$
]

---

# Generated .darkblue[Black Box] Function

.more-more-condensed[
- Quantum computation: .darkblue[circuit of quantum logic gates<br/>$\Rightarrow$ circuit must be generated] dependent on the concrete problem instance, no general circuit to solve all instances of a problem
- Sketch of algo:
.smaller-font[
<table>
<tr>
<td style="width:65%">1. Determine Separators and Permutation</td>
<td>$O(m+n)$</td>
</tr>
<tr>
<td>2. Check Validity of Separators</td>
<td>$O(m)$</td>
</tr>
<tr>
<td>3. $\forall i$: Determine lengths of $i$-th transaction in permutation</td>
<td>$O(n\cdot log_2(n))$ with decision tree over transaction number</td>
</tr>
<tr>
<td>4. Check: Which separator configuration? For current case:</td>
<td>$O(n)$</td>
</tr>
<tr>
<td>  4a. determine total runtime of core and check if it's below given limit</td>
<td>$O(n)$</td>
</tr>
<tr>
<td>  4a. determine start and end times of conflicting transactions</td>
<td>$O(n\cdot log_2(min(n,c))+c)$ with decision tree over conflicting transactions (for $n>>c$) or transaction numbers (for $c>>n$)</td>
</tr>
<tr>
<td>5. Check: Do conflicting transactions overlap?</td>
<td>$O(c)$</td>
</tr>
<tr>
<td></td>
<td>.darkblue[$\sum:O(n\cdot log_2(n)+c)$]</td>
</tr>
</table>
]]

---

# .darkblue[Complexity Analysis]

.more-more-condensed.smaller-font[
<table class="result tablepadding">
<tr>
<th>Approach</th>
<th>CPU</th>
<th>Quantum Computer</th>
<th>Quantum Annealing</th>
</tr>
<tr>
<th>Preprocessing</th>
<td class="textcenter">$O(1)$</td>
<td class="textcenter">$O(n^2\cdot c)$</td>
<td class="textcenter">$O(m\cdot R^2\cdot(c\cdot m + n^2))$</td>
</tr>
<tr>
<th>Execution</th>
<td class="textcenter">$O(\frac{(m+n-1)!}{(m-1)!}\cdot (n+c))$</td>
<td class="textcenter">$O(\sqrt{\frac{n!\cdot n^m}{k}}\cdot(n\cdot log_2(n)+c))$</td>
<td class="textcenter">$O(1)$</td>
</tr>
<tr>
<th>Space</th>
<td class="textcenter">$O(n+m+c)$</td>
<td class="textcenter">$O((n+m)\cdot log_2(n))$</td>
<td class="textcenter">$O(m\cdot R^2\cdot(c\cdot m + n^2))$</td>
</tr>
<tr>
<th>Code</th>
<td class="textcenter">$O(1)$</td>
<td class="textcenter">$O(n^2\cdot c)$</td>
<td class="textcenter">$O(m\cdot R^2\cdot(c\cdot m + n^2))$</td>
</tr>
</table>
<div class="padding-top">.bold[$m$:] number of machines .bold[$n$:] number of transactions .bold[$c$:] number of conflicts .bold[$R$:] max. runtime .bold[$k$:] number of solutions</div>
]

---

# .darkblue[Number of Solutions]

<table>
<tr>
<td class="halfwidth">
<div class="textcenter">
<img src="../../img/ideas21/NrOfSolutions.jpg" align="top" class="fullwidth"></img>
</div>
</td>
<td>
.more-more-condensed.smaller-font[
<table class="result tablepadding">
<tr>
<th></th>
<th>$m=2$</th>
<th>$m=4$</th>
</tr>
<tr>
<th>$N$</th>
<td style="text-align:right">8,589,934,592</td>
<td style="text-align:right">2,199,023,255,552</td>
</tr>
<tr>
<th>$k$</th>
<td style="text-align:right">48,384,000</td>
<td style="text-align:right">559,872</td>
</tr>
<tr>
<th>$k$ for<br/>$\leq 1.25\cdot R_{opt}$</th>
<td style="text-align:right">1,472,567,040</td>
<td style="text-align:right">2,047,306,752</td>
</tr>
</table>
]
</td>
</tr>
</table>

---

# .darkblue[Summary & Conclusions]

.more-more-condensed[
- .darkblue[Scheduling transactions as jobshop problem with] additionally considering .darkblue[blocking transactions]
  - .darkblue[Hard combinatorial optimization problem<br/>$\Rightarrow$ hardware acceleration]
- .darkblue[Enumeration of all possible transaction schedules] for finding an optimal one
  - Grover's search offers approx. .darkblue[quadratic speedup] to approach on traditional computer
- .darkblue[Estimation of number of solutions for] a further .darkblue[speedup]
  - Estimation of speedup for suboptimal solutions being a guaranteed factor away from optimal solution
- .darkblue[Code Generator] available at .small-font[<a href="https://github.com/luposdate/OptimizingTransactionSchedulesWithSilq" target="_blank">https://github.com/luposdate/OptimizingTransactionSchedulesWithSilq</a>]
- .darkblue[Future Work]
  - .darkblue[Quantum computing of other database problems] 
]